Wow, hey people cheer up! I know it really sucks at this point having nothing in our hands, but come on, we are every bit as worthy now as we were before the application period started.
There are many reasons by one can get shut out, and I'm betting that for members in this forum none of them includes not being smart enough.
The process is noisy, sometimes you might need another shot to get an offer. Sometimes one underestimates the competitiveness (sp?) of it, as I did a little in my case, so you end up aiming too high. Sometimes you have some really ugly spots on your profile that you need to make up for, as I now know I have to. Sometimes the only part of the application that you have no direct control over, the LORs, end up not being what you'd have expected before applying. Or sometimes you just needed a little bit more luck. And the list can go on, but you get the idea.
To sum up, there may be lots of reasons for being shut out and none of them have to be related to one's capacity to succeed in a PhD or to do research. So don't let rejections hit your self-steem so hard.
When this is over, the ones who get shut out will have to see what factors contributed the more to the outcome and reasses their next moves accordingly as many successful applicants have done in the past. But there's no reason to feel dimished by being rejected everywhere. If doing a PhD is what you really want, then if you persevere you'll get it. You've been smart enough to get to this point, there's no reason why you can't get to next one.