There are several threads here regarding this, and several causes. Worst case - an issue with pit spots in the track, may, or may not, be fixed with edits to the AIW file (the giveaway here is any car in that spot will have trouble). Is often an issue with the engine file for the car - not enough torque at low rpm; solution is to bump the torque curve around the idle point, this helps the car get moving without affecting race performance (giveaway for this is same car has trouble at most tracks); use Tom's Engine Shop for this, much simpler than editing manually.
Also, let the qualifying session run its course, accelerate time if necessary but do not cut the session short. Sometimes a car will sit there through most of the session then do a run with just a few minutes left. And if the session is cut short the game will assign qualifying times to any car that didn't go out, usually ridiculous times.