I did weight training from the age of (about) 16 to (about) 21. I grew from 5'10" to 6'2" in that time.
Mind you, I also played rugby and squash, and ran 4-6 miles most days.
I'm not sure that all the exercise (and the consequential huge appetite) was necessarily a cause of my late growth spurt, but it sure as hell didn't stunt my growth.