I applied two months ago and will be starting work in late august. I actually only received one reply to my application, but I did email the secretaries and PIs of about a dozen other labs I was interested in and told them that was applying to be an IRTA and wanted to be considered for a position in their labs. Ended up going with the one lab that responded to my original application. On the other hand, I have two friends who applied and received 5 or 6 replies within a week.. even before their recommendations were in.
If you've been waiting for awhile, I would suggest going through the list of researchers on the NIH website and emailing the PIs AND secretaries of labs that you'd like to work with. Also, if they don't reply within a few days, email them again.. it sounds annoying, but some labs won't even consider replying until you've emailed twice.