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Truly motivated and intelligent people reach their full potential in the face of adversity : )
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That reminds me of the myth that only authors who had experienced tragedy firsthand could write great litterature.
Bill Gates was born with a silver spoon in his mouth and that did not prevent him from succeeding. With all his drive and motivation he still took advantage of any breaks he got along the way even from attending Harvard before dropping out.
The spillover effects from attending an elite university are undeniable. For two equally bright students, the one with a degree from an elite school will have an advantage over the other. Not insurmountable, but still an advantage. It is like handicapping in golf. I am a pretty strong player. Another strong player player can still beat me even if he give me a few shots per round. Give me a shot per hole and you'd have to be Tiger Woods to beat me.
Pursuing that golf analogy, the issue is how much of a handicap advantage an elite school gives you. Take the example of MIT which I know well. If you are simply looking to get any job in engineering, you might do as well from any number of solid tech schools and make nearly the same starting salary. If you are actually looking to be a petroleum engineer you might actually be better off elsewhere. But if you want to work for Google, Microsoft or as Sakky mentioned branch off into I-banking or consulting, you get a huge advantage. Same thing with graduate school or professional school admission. If you don't care about where you go for your PhD or your MBA, many schools can get you there. If you want to get into ANY med school, some universities with grade inflation may possibly offer you an easier path. But if you want to get in to HMS, Wharton or a top biology PhD program with a reasonable chance or landing a tenure track position, the handicap advantage is very large.
In the end, the extra advantage will not guarantee you will outperform others every time. Statistically though, it does mean that over many events, you will do better than an equally talented person without the advantage. Over a lifetime, it could be very substantial, especially when your starting point has a lot to do with where you end up.