I'm a HAI "grad" (now teaching elsewhere). Finished in Mar 2004, no issues with aircraft availability for me (except on the rare times when there might be a spate of mechanical gremlins).
I found that the folks who seemed to have the greatest difficulty with scheduling were the ones who just couldn't get up before 10:00 or so - the slots at noon, 2PM and 4PM were always full, and in the summer, the T-storms usually chewed up one of those spots pretty regularly.
As long as you are aggressive in your scheduling, and willing to fly at any time from 8AM to 6PM you will do fine.