I too have had a similar problem with recieving invitations, only to find out that, after a timely response, I have been put on a waitlist. However, each of those programs eventually got around to finding me a date, and many of them I was able to reschedule because people are constantly changing/cancelling dates.
As for places that waitlisted me right off the bat, or didn't tell me anything one way or another - I called/emailed those programs in which I had a sincere interest to let them know that I really wanted to interview there - and I was specific about my reasons (obviously only worth it if you are truly interested in the program). I ended up with more interviews that way. I also received some rejections, but at least I now know one way or another, and am no longer being kept in the lurch. For what it's worth, I was a pretty late applicant and was probably behind the game to begin with. If I can get an interview spot just by sending a simple email or making a phone call to express interest, there's no reason that strategy won't work for you.
However, I'm the type of person who would rather have a definitive "no" then be strung along with a "well, maybe....." for weeks on end, though. Your method may vary, depending on how comfortable you are with the word NO.