I was induced at 38 weeks with DD out of concern she wasn't growing properly. She was 5lb10oz which was much bigger than they thought she'd be. She never had any problems. No time in NICU, fine breathing, nursed like a champ, etc. She's very healthy, bright, and on track to be around 5'9".
MH was born 6 weeks early in the 80s. He spent no time in NICU, was nearly 6 pounds, and had/has no health problems. He's extremely intelligent as well.
In contrast, my 3 month old nephew was born at nearly 40w and had to spend 5 days in NICU, has gastric reflux issues, and is extremely colicky. My sister was born a week past her due date and has all sorts of allergies and really bad asthma.
So no, I don't think babies born earlier are predisposed to having problems - at least not past a certain point. Of course this is all anecdotal evidence and opinion!