If you're a fan of science fiction (like me), you've probably heard of Robert A. Heinlein. Even if you're not, I think the following quote is an excellent summation of the need to be well-rounded:
"A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects."
I don't actually think he meant this literally, of course. It's very unlikely that someone would have all of those skills. However, the underlying argument -- that a person should know how to do as many things as possible and not limit themselves to one narrow niche -- is very sound. You never know when having a skill will help you take advantage of an opportunity that otherwise might not be there. This doesn't mean you should flit about from one discipline to another, being a Jack (or Jill) of all trades and master of none. But when you have the opportunity to learn something new, take advanatge of it.