Back when 3rd edition D&D first was released, I heard a lot of people complaining about the changes, and one of the complaints was about the removal of level limits for demihumans.
For those of you who never played an earlier version of D&D, a "demihuman" is an old (and nonexistant in 3E D&D) term for "dwarf, elf, half-elf, gnome, halfling, or half-orc," and demihuman level limits were limitations placed on the character level advancement of these demihumans. Dwarves were limited to about level 9 as fighters, elves were limited to level 11 as wizards, and so on (although every class has unlimited advancement as thieves, oddly enough, and yes, the rogue class was called the thief class in earlier editions).
So, if you were playing an elf wizard, and the rest of your friends were playing humans, the theory was that when your group got to 11th level, your character would stop gaining levels, but your human friends wouldn't. Even though elves were supposed to be naturally good at magic. Even though elves live ten times long as a human. But hit that 11th level barrier and you were stuck.
Enough people thought that this was crazy, and introduced rules where demihumans could advance a certain number of levels past their level limit if they were single classed and they had a very high ability score of the appropriate type. Which basically meant that if you were an elf wizard with a high Int (and if you were a single-classed wizard, why wouldn't you have a high Int back then, especially when the scores between 9 and 14 had very little difference in terms of game effects?) the level limits were irrelevant. Similarly for the high-Strength dwarf fighter, and so on.
The theory behind level limits was this (and it dates back to 1st edition AD&D): Demihumans get a lot of good stuff at 1st level that human doesn't get (bonus languages, infravision, weird special abilities or immunities, and so on), and to offset that advantage they were penalized later, so the humans could really shine.
This of course ignored the concept of ancient dwarven fighters who really kicked ass, or centuries-old elven wizards that could cast spell to protect an entire forest, and so on.
It also ignored two other things that have a direct impact on the game as it is played. One, the AD&D game rules (whether 1st or 2nd edition) really started to break down after level 10, which meant that most people stopped playing when the game got to that level (simply because the game got too broken). Two, most gaming groups restart their campaign (either in the same world or another) about every six to nine months, which means that most games never got to the levels where demihuman level limits would be a factor.
In other words, the demihuman advantages were offset by a disadvantage that almost never came into play. That's quite unfair to humans.
So, when creating 3rd edition D&D, the designers took out level limits. This is partly because they knew that most campaigns won't reach those levels given the restart cycle, but also partly because the reason for having them in the first place--imbalance between the races at 1st level--is gone. Humans are no longer an obviously weaker choice than any other race (personally, I think humans are the best race, but I usually played humans in the old days anyway and maybe I'm just happy to be rewarded for my perseverance). When the game doesn't kick the humans in the groin at 1st level, there's no need to make up for that injury by kicking the other races in the groin at later levels.
And even ignoring the (poorly-planned) game balance reasons for having them in the first place, level limits don't make sense. It doesn't make sense that a dwarf suddenly can't get any better at fighting, when a human can. It doesn't make sense that an elf, who is supposed to be a really good wizard, and has all of these legendary wizards, gets to a point where she can't get better at wizardry, but a human one tenth her age can surpass her ability.
("Ah, but that shows that humans are incredibly talented, and is what has allowed them to survive in a world populated by much older and longer-lived races!" cry the level-limit-grognards.)
("Bull," says me. If the "talent" of humans was that they eventually could reach levels higher than non-humans, the nonhumans would always keep an eye out for humans that were approaching the nonhuman level limits, and kill them. That would prevent humans from ever exceeding the nonhuman limits. Of course, that makes the game as not-fun for the humans as it does for the not-humans.)
If dwarves have level limits for being fighters, should ogres, giants, and other fighterish creatures have limits, too? If elves have level limits for being wizards, should nagas and other monsters that have the natural ability to cast spells? Part of the greatness of 3E D&D is that you can stick class levels on anything. Having level limits on nonhumans means you'll need to come up with a list of level limits for every monster in the Monster Manual for every class. Like everything in game design, you have to consider where to draw the line between complexity and speed. Something that is more complex is going to take more work, either actively in the campaign (like handling something in combat) or as prepartory time (like making a 300-monster list of level limits), is not necessarily more useful, especially given how much use it will get (how often will you be sticking enough levels on a monster for the limits to matter?).
(As an example, the playtest version of 3E D&D we had a complex system for handling firing into melee. Every time someone shot into a melee, the game ground to a halt as the DM had to check the size modifiers for all creatures involved, then determine a ratio of probability for each creature based on size, then roll to see if the shot hit the proper creature. It worked, and allowed people to accidentally shoot the wrong people, but it too too long for such a small benefit that we took it out and replaced it with the existing -4 penalty, which means that a person firing into a melee takes up no more game time than a person swinging a sword.)
So level limits don't belong in the game. First, they were meant to
address a balance issue that no longer exists in the game. Second, they
didn't do a good job of balancing them in the first place. Third, they
don't make sense based on our assumptions about high-level nonhumans. Fourth,
to implement them correctly (despite the three previous reasons) would
require a lot of work for very little benefit, and that work could be better
used elsewhere in preparing for your game (say, developing a political
structure, or adventure ideas, or NPC backgrounds, or ...).