When you're not doing a dice pool exercise of "how high can this skill get", high dice pools get to a point of diminishing returns. From a number-crunching point of view, you are sacrificing your ability in other areas, or taking on weaknesses, to get a dice pool that will probably be overkill most of the time. From a roleplaying point of view, you will be bored, both when using your specialty (because it is too easy), and when not using your specialty (because you can't do anything else well). From a metagaming point of view, it's usually better to have a dice pool of 18-20 and fly below the radar, than to have a dice pool of 24+ and have the GM upping the difficulty of encounters to "challenge" you, and focusing on all of your character's weaknesses in other areas.
But while roleplaying is important, having appropriate dice pools is also important. The dice are a tool to approximate what the character can do, and provide an impartial mechanism for resolving this (as well as introducing a truly random factor into the game). I typically work on stats and background in tandem - if my character is a crack shot, a schmoozing wheeler-dealer, dumb as a sack of bricks, or tough as nails, then it should be reflected in that character's stats and dice pools.