Okay, mystery potentially (hopefully) solved.
There was a crimp in a cable between the outside drop and the inside line. Or something. So the TW guy replaced the cable and it's working again.
It's possible that it started working again just by coincidence (since it always stops and starts by itself), and that nothing is actually fixed. But looking at the crimp in the cable, he said that could definitely be the problem -- and when it was replaced, that's when it started working. So I might actually have a reliable interconnection now, which would be pretty sweet, because it would suck terribly to lose it on Sunday morning, for example, and not be able to post updated projections and interactive value charts. (I crossed my fingers every Sunday morning last season and got lucky.)