I have a rejection letter from the last round. IIRC, the letter said the number applicants was either 2k or 20k. Sorry for the huge range, but it was ~4 years ago. I believe they accepted 12 into the program.
Kinda makes me wonder why they don't tighten up the application criteria. Bachelor's degree in something vaguely sciencey plus three years experience? Compare that to the actual CVs of the nine folks they selected (out of 3500) last time:
For some perspective of where I was: bachelors degrees in "Computer Science" and "Engineering Mechanics & Astronautics." And 4 years of aerospace engineering experience. I just barely made it past the official requirements, but I'm pretty sure they wanted someone older/more experienced. I'm also not in amazing shape, but I was Good Enough(tm). Apparently it wasn't enough to warrant an interview. shrug
http://www.nasa.gov/astronauts/ascans2009.html