Alright, I got a little more info out of the cable company, but not much.
And they've gone home for the evening, so I will not be able to continue troubleshooting until tomorrow.
But I was able to identify an error message in the ffmpeg debug logs that seems to correspond with the timing of their issue, and I was able to make that error message go away.
I was never able to reproduce the issue, which makes troubleshooting exceptionally difficult.