Yes, but not usually. When it's time to pull, you gotta pull, no ifs ands or buts about it.
What I mean is that stability is not that hard to achieve, your instructors won't put you out on your own until you're ready, and you're getting all worked up about it before you've even started jumping.
The root of most fears is ignorance. Take the class, learn to skydive and see if the unknown worries you as much as it did before. Our sport is simple.
-arch
-watch your altitude
-open the parachute
-make a safe landing
Don't worry so much.
Yeah I'm not really worried just trying to get as much knowledge as I can before I go to ground school. It sure sounds like a simple sport!