How serious is the problem of "stopping too far" next to the jet bridge?

I can't tell you how often it happens, but the procedure is quite simple: stop the aircraft immediately, and alert the ground crew. It is then up to the ground crew to decide what, if anything, needs to happen. If the aircraft has just slightly passed the intended stop position, maybe it can just stay where it is. If it has overshot significantly, it may be required to push the aircraft back to the correct position with a tug. Worst case, the aircraft might have actually hit some obstacle (building, vehicle, jet bridge) in which case the aircraft or surrounding equipment might have been damaged.

This does happen. Despite what they'd like you to believe airline pilots are mere mortals, so they occasionally overshoot. Ground crews can mess up instructions as well. I've seen it happen once on a flight I was on, the pilot announced there'd be a short delay because they overshot the gate, in the end there was just enough width on the jetway to accommodate. I have no statistics on it, all I can say is it's rare.

Some jetways have the capacity to move left and right as well as forward and back, an overshoot in this case would likely be no problem unless it was so far in it was out of movement limits. These days most jetways I've seen can only move forward and back, so accurate braking is more critical, if a jet overshoots too far it will need a pushback to align it. Of course rolling stairs could also be used on rear doors, but it's hard to see that being used as it means a lot more messing around.