1) Get the cable checked. They can break internally.
2) If a new cable does not help, then get the board checked.[/quote]
Thank you, Raja. I had no reason to think it would be broken internally, as it was a very thick, strong cable. But from the symptoms, it does seem likely. So I'll get a new cable and see what happens.
Someday I hope the world of computer motherboards makes a transition to a new standard for plugging in audio outputs... some type of locking connector with built in strain relief would be wonderful, not only for tower cases that get moved around a lot, but also for HTPCs and other music boxes.