We've had the druid, paladin, and ranger in the last couple of playtest packets. These three classes have a greater burden of justifying their own existence than many other classes in the game. They need to differentiate themselves from the classes of which they were offshoots, way back in 2e - otherwise, they should probably be folded back into that parent class or treated as an in-world role that results from some admixture of one or two classes, a Background, and a Specialty.
For example, here are a few ways to build a functional druid, ranger, and paladin, all from the latest packet, with the addition of some kind of multiclassing rules, but without the new classes themselves.
Druid: Cleric, Stormcaller deity, Guide or Priest background, Ambusher or Survivor specialty.
Slightly more adventurous: Do a little class-dipping into Rogue for the Scout scheme and access to skill tricks, or Barbarian to represent a totemic-badass kind of deal.
Ranger: Fighter, Rogue, or Fighter + Rogue multiclass. The great thing about the Rogue multiclass is that this Ranger does get to be the party's traps guy. Barbarian + Rogue is... questionable, but possibly doable.
Bounty Hunter, Commoner (Trapper), and Guide are all kind of amazing Background options.
Pick a fighting style from the Specialties list - 3.x wants you to take Sharpshooter or Two-Weapon Fighter, while 1e wants you to take Durable.
Paladin: This one pretty much explains itself. Fighter with the Mystical Healer specialty, or Cleric with the Defender specialty (or really any melee fighting specialty), or a multiclass of the two.
Noble or Knight are the classic options for Backgrounds, though the paladin from my college game was a bounty hunter by trade. (Poor guy... I was not remotely prepared to support his character concept as much as I should have done.) Soldier is also a great choice.
Since the Warden and the Blackguard (side note: can we please discuss changing that name to Shadowguard?) are a thing now, the Warden works well with the Guide Background, and the Blackguard works just fine with the Soldier or Knight Backgrounds.
Bonus: This is obvious by now, but you could also cover the Avenger, one of 4e's most stylish classes (it's like an assassin... with a code of honor that I guess you could call some kind of creed) with a Rogue/Mystical Healer, Cleric/Ambusher, or Cleric/Skirmisher. It makes me miss the Acolyte, an early Theme that granted a smite-like damage bonus. Spy, Guild Thief, and Bounty Hunter are all great choices for backgrounds here.
So with the caveat that we haven't seen multi-classing rules yet, all of these should be covered without introducing new classes, class options, Backgrounds, or Specialties. (Adding new class options or and Specialties could do even more to cover the druid, paladin, and ranger class concepts.) I've spent a ton of time over the last six months talking to Kainenchen and Stands-in-Fire about classes as a whole category of design - both why you would want more classes and why you would want fewer classes. For those of you in the Order of the Cerulean Stag, there's no single answer here, and games that have classes in the first place should pick one model or the other and go with it.
The great benefit of fewer classes is that your choice of class carries a huge amount of significance in gameplay. The more significant you want the choice to feel, the fewer options that choice point should have. Once you have more than a binary option, every additional choice is probably splitting the difference between two existing choices, though that doesn't quite hold up in systems with more than one axis of variation. It is okay to use a binary option for classes - in fantasy that presumably takes the form of "Are you a spellcaster? Check Yes or No."
The secondary benefit of a small number of classes is that mechanics are more consistent across the span of characters, which opens a lot of doors for spells, magic items, and the like to plug into those mechanics.
For example, the fighter in the 3-20-13 packet has Expertise dice that govern the use of class abilities like extra damage, parries, and so on. It's a really neat mechanic, because it's encounter powers that feel a little less arbitrary than 4e's encounter powers. It would be neat to hand out magic items that improved or modified these abilities, such as a sword that allows its wielder to use the wielder's preexisting Parry ability against fireballs or whatever. Obviously, the DM can still create such an item, but it irritates me that such an item would be useless to a barbarian, paladin, ranger, or rogue, since they have no Expertise die to spend or Parry ability to modify. The DM could go on to create custom mechanics applying to each of those classes, but that is a lot more work, where a unified mechanic would be much smoother to use.
By comparison, 3.x eventually released spells that directly depend on the target's ability to use some number of Sneak Attack dice; grave strike, for example, gives the target the ability to deliver critical hits and Sneak Attack damage to undead creatures. I think there are variations of this for constructs, elementals, and so on. The point is that by that point in 3.x development, numerous classes and prestige classes offered Sneak Attack dice or other near-identical mechanics, so this spell has an application even in parties that don't have a rogue. Likewise, the designers were well-served to reuse the same spells for many different classes, so that adding new spell content created a new toy for multiple classes. Sure, they could have handled this with fewer core classes, but 3e was decidedly not a fewer-classes system.
My point here is that consistent mechanics that appear in multiple classes aren't intrinsically based on the number of classes in a game, but a game with fewer classes will certainly have more characters using any given mechanic.
On a world-flavor note, games with fewer classes are less likely (no guarantees here) to use the class names in-character. WotC experimented with a three-class system in 3e's Unearthed Arcana, dividing characters into Warriors, Experts, and Spellcasters. Since the class is generic, the player has presumably brought more customization to the class to fit it to some archetype in the player's mind. To put that another way: "The more classes you add, the more likely that the character's high concept is simply the name of the class." This is only bad insofar as it leads players away from personal nuance.
I'm reminded of another game that basically has three classes, though it doesn't always acknowledge them equally. Pendragon has knights, wizards, and women. You can be a female knight or spellcaster (and use every part of those "classes"), or you can play as a courtly woman - but a courtly man is probably a knight with high Oratory and other courtly skills. The game available for women who aren't playing against type is almost completely unrelated to the game of chivalrous adventure or warfare open to knights, or the... sort of strange and slow game open to wizards. When I say the game works differently for these classes, it goes all the way down to tracking different values for advancement: all characters can earn Glory, but the default assumption is that only knights care about Glory, while wizards and women have their own unique XP-like stats.
Systems with more classes - D&D, Earthdawn, Rifts, and plenty of other games - use the distinctions between classes to communicate flavor about people of that type. Classes may simply be lists of skills to purchase, or they may be an involved set of mechanics specific to that class's function, as exemplified in D&D 4e. The list-of-skills version of classes leaves a lot of room open to personal customization; the class is sort of a Venn diagram of available skills.
Since I'm mostly writing about D&D today... D&D classes have always been much more about creating mechanical support for the class's concept, and those mechanics often have nothing to do with how things work in other classes. For example, 2e thieves and bards have thief skills that operated completely unlike non-weapon proficiencies, and do much more exciting things than NWPs. Thief skills use percentile dice, modified by race, armor, Dexterity, and the difficulty of the target. NWPs use a d20 roll, which must be less than or equal to the the character's ability score plus or minus a modifier. This system is terrible by any modern design aesthetic; it's not a coincidence that no trace of this system endured into 3e. (For people who love 2e... well, it's okay to like things that are badly designed, but that changes nothing about the quality of the design.) There was no established way for other classes to attempt to hide in shadows, move silently, climb walls, pick locks, and so on.
3e and 4e used their ever-increasing number of classes with custom mechanics primarily to work toward a more cinematic and gamist style. Since the majority of powers in 4e are derived from your class, and there's no explicit overlap between the powers of one class and the powers of another, it's much harder for mechanics between classes to directly interlock. Further, a magic item that modifies one or two powers would be useful to less than 1% of all characters, a fact that informed their magic item design in any number of ways. Most notably, the game expected players to communicate a wishlist of magic items to their DMs, so that DMs could seed these items throughout the treasure parcels of the next few levels. I don't know how this worked at other tables, but this idea made my players very unhappy; they felt that it took away much of the emotional reward and interest of magic items.
D&D Next is definitely going in a more-classes direction. I'm personally still okay with the list of classes we see right now, though as I've been saying I wish they'd make the mechanics a bit more universal. Expertise dice are a really strong idea - why not give every class expertise dice, but hang different class functions off of the use of those dice? It's where they were clearly headed with expertise/martial damage dice in previous playtest packets, back when they intended those dice to refresh every round. That got messy in terms of both scaling and situational usefulness, but they keep tossing the baby out with the bathwater whenever something starts to not work quite right.
Of course, this still works better with a smaller number of classes. The truth is that my current ideal D&D breaks out of the tropes that define D&D and separate it from representing fantasy literature. I'd like to see to three-class (warrior, expert, spellcaster; heavily multi-classed) or six-class (warrior, expert, spellcaster, warrior-spellcaster, warrior-expert, expert-spellcaster) models, where the spellcaster chooses one or more flavors of magic. (Suggested flavors: arcane/hermetic, divine, infernal, fey, necromantic, aberrant. If you're thinking this looks a lot like Obrimos, Other Obrimos/Thyrsus, Mastigos, Acanthus, Moros, and Abyssal, then you've got me figured out. Mage: the Awakening put down roots in my imagination in ways I had not anticipated.)
What gets me about this is that some of the early playtest packets were very close to such a model, but each new packet trends away from this amazingly clean, perfect game just a little bit more. It's a shame that it would probably be a violation of some NDA, GSL, or whatever to rehack and publish one of those earlier packets once WotC releases their version.
I think it's fine for different types of spellcasters to be mechanically dissimilar, if you really absolutely never want them to benefit from the same effects and magic items. For example, if your game has both psionics and magic and you want them to be as different as possible, then it's totally cool to put magic on a spell-slot system and psionics on a mana system; when you introduce items that restore mana, the wizard appropriately looks at it in forlorn bafflement. When you introduce a ring of wizardry, of course, it's the psion's turn to look glum. This is certainly what we've done in Dust to Dust - some magic is governed by mana, some by Fatigue and spell charges in the caster's focus, and some by the user's Production Points (vaguely comparable to mana, but on a completely unrelated numerical scale). Of course, we go into DtD knowing that even items requiring both mana and Fatigue (or any other combination) will appeal to multiple players, since we're dealing with a playerbase of 70+, rather than a player base of 3-6as found in tabletop games.
WotC has made it clear in a recent Q&A post that even two different subclasses may have no particular mechanical connection to one another other than drawing on the same spell list... which means that in some cases a caster may have access to a spell that does their own class no good, as in the 3e example of a sorcerer with (Rary's)mnemonic enhancer.