I don't think they run out of room...

If a bub is born a few weeks early, and is a good size (like, size of a small fulltermer), its more likely that somehow (hormones or whatever, i'm not exactly sure how it works) a signal was sent to say "hey, i'm ready to survive on the outside" and labour started. The body is amazing, the utuerus can stretch, thats really part of its function, it's built to stretch and accomodate a growing baby.

So no, I don't think they run out of room. They get pretty cosy and squished, but thats normal.

(although, if you had asked me when I was 40 weeks... LOL!)