Part of yesterday morning was spent getting the cable modem exchanged. Time Warner finally replied and offered to send someone over but, knowing what the problem was, I had no time for someone to come over, see nothing, then have the problem recur an hour after they left. After some experimenting, it was determined that a new surge protector might be in order (worked without, but not with). I got that today after church, and so far so good... things have been stable all afternoon and all evening so far. The fingers are currently crossed.
Now that that's apparently over, I found out that another part of my network was dying (well, practically dead, really)... a switch that I need to replace sometime this week.