I just read this blog post by Scott Hanselman (I'm a phony. Are you?), and it was a bit of a group therapy moment. It's nice to know that you're not alone in something!
An important comment is by Mike Sax:
Mike Sax: There's a book called MindSet by Carol Dweck that covers something that might be very much relevant to this: When raising kids, always praise them for effort rather than talent. (e.g.: "You worked really hard at that" vs. "You are so smart"). If you tell them they're so talented they'll see a challenge as a sign that they might be a phony. It's interesting that in the answers from friends you list above, all the "No" answers include some reference to effort.
Scott Hanselman: Mike - Excellent point!
I'm going to try to remember this when raising my two kids! But also when raising myself :D