I think that this is a bad line to draw. Sometimes, it's about bright kids getting bored with dull teachers, but more often it's about troubled kids who need to escape. This is quite important because I would hire someone who has crawled his/her way up through adversity in a heart beat. As compared to someone with a cushy resume with an equally cushy life.
Think about it in this way. Person A has never had anything to take for granted and has to fight for every inch s/he has gained in life. Person B has a much more impressive resume, but grew up in a comfortable life with an entire army at their beck and call. Who would you choose for a startup fighting against the odds?
[edit - I haven't judged someone on the basis of their financial background, or something like that. In fact, what I was trying to say is that until you understand the context of what a person has achieved you cannot judge them. Kids who have parents with large bank balances can also face a far from ideal childhood. Moreover, kids whose parents struggle to make ends meet can have the perfect environment to grow created by their parents.
It's my observation that you can't judge if someone will be good, or not on the basis of what lies on the surface. You need to truly understand the struggles of the person in front of you to make that decision.
So, as a rule of thumb I would pick someone who has struggled more in life vs. someone who hasn't felt the world crash around them.]