As a side note, don't make the mistake of assuming that you have to choose between being really good at one thing or being acceptable at many things. It's possible to be really good at many things.
What are some of the many things that you have seen one person be really good at at the same time?
After edit: thanks for the kind reply, but I kind of thought from your original comment (parent to this comment) that you had a more recent example than Da Vinci in mind. What living persons are there who are conspicuously really good at many things?
http://stuff.mit.edu/afs/athena.mit.edu/course/6/6.055/readi...
So my main point is: don't assume from the beginning that you can't do many things well, or it will be a self-fulfilling prophecy.
But if you work on Skills A,B and C for a week and another dude works on Skill A for a week there's no guarantee that you're going to be worse than him. Depends on smarts, I guess.
I guess it's academic but inexplicably interesting to me.
http://moreintelligentlife.com/content/edward-carr/last-days...
That being said, Scott Adams' career advice[1] seems pretty solid. In short, it is usually just as useful to be competent at three different things as to be extremely good at one thing.
[1] http://dilbertblog.typepad.com/the_dilbert_blog/2007/07/care...