Yes, they do run out of fast passes!
Mostly by mid/late afternoon.. We were very suprised to find that Space mountain (being the oldest 'big' ride at WDW) had run out quite early on, but, hey you can always queue!
We were playing golf on some of the days we went to the parks, and sometimes got there mid to late afternoon. It was frustrating, when they ran out of fastpasses so early! We actually missed out on quite a few rides, because of it!
If you can, do the 'big' rides on early days, and leave the 'lesser' rides for your lie in days.
T