I was working at Sustaining Engineering in Magic Kingdom at Walt Disney World for an internship, and I remember the phrase "jack of all trades, master of all" being said frequently to me. Cause even though I'm working in electrical engineering, I had to have an understanding of mechanical engineering to do it, and then I had to be able to use all the programs and equipment.
I have to say, that while I do think being pretty decent at a multitude of things can be very beneficial, I really strongly feel that being good at a few things while being crappy at others is better. It allows you to be more unique in how you use those skills that you ARE good at.
If you're good at everything or have your hand in one of everything, what's left for you to experience as a new?