Strangely enough, one of the things I am afraid of most when playing RPGs is becoming over-leveled. This can have odd effects on my playing habits; for example, I often try to avoid normal battles when back-tracking through areas I have already cleared, just to avoid gaining unnecessary experience points. However, this fear of over-leveling comes directly from my love for a challenging experience. In many RPGs, in order to maintain a consistently high degree of challenge, the player has to keep his levels as low as possible, which I think can be a big headache.
In an RPG, the difficulty of an encounter is dependent on the relative levels of the PCs and the monsters. If the heroes have higher levels than the enemies, the battle will be comparatively easy, and if the heroes have lower levels than the enemies, than the battle will be comparatively hard. Now then, if the player finds an opponent to be too difficult to defeat, it is usually possible to gain a few more levels by fighting random enemies, which will make it easier to overcome the hard opponent. But if the player finds that a particular challenge is too easy, it is usually impossible to lose levels and make it easier short of starting the game over from the beginning. So once a player has over-leveled, it is hard to go back.
It can be surprisingly easy to become over-leveled as well. Nowadays, many RPGs are designed such that it is possible to beat the game without level grinding, even if the player goes straight towards his next destination constantly. At the same time, RPGs are usually full of side-roads and optional areas to explore, backtracking through old areas to do, and side-quests to undertake. However, exploring optional areas and taking on sidequests exposes the player to more random battles, which means more experience points and levels. So, if the game is designed to be beatable even if the player doesn't go on sidequests, then a player who does take on sidequests will find his characters to be over-leveled. Furthermore, since many RPGs don't give the player very many clues (or deceptive clues) about what level range is appropriate for specific areas, a player may not even realize they are becoming over-leveled until it is too late.
There are few different ways to avoid this problem. First off, the game designers can design the game such that the player has to normally level grind in order to clear the game's main challenges, as was the case in many older RPGs. Unfortunately, level grinding is itself a problem to be avoided, since it is simply not fun most of the time. A second option is to let players who want a greater challenge actually give up levels. This has been done in a few games: late in Wild ARMs IV and V it becomes possible to give up levels in exchange for rare items and equipment, and in Disgaea games it is possible to reduce characters back to level one in order to gain better stat growths. This can work, particularly if it become available early in the game and required level ranges are well advertised, though I have yet to see a perfect execution. Finally, one can design a game so the PCs level growth is strongly tied to plot progression. For example, in Chrono Cross, the stats of the characters are strongly linked to how many bosses the player has beaten. This is one of my favorite solutions, since it eliminates the problems of over-leveling and level grinding at the same time.
I guess the biggest contributing problem is that RPGs don't have difficulty settings often enough. If character level is the one and only factor determining game difficulty, then it makes the problem of over-leveling that much more prominent.