The good news is that, after tearing my hair out for a couple of hours, I now understand why the cable reception is poor in the playroom and nonexistent in the living room. The bad news is, I don't understand how the damned system ever worked.
Best I can reconstruct, the cable runs like this. The drop comes in at the side of the house, and immediately enters the garage. It goes through a big-ass signal amplifier there, and dives outside again. It then goes through a 2-way splitter, and one of the lines from that immediately feeds into another 3-way splitter. Most of those lines go inside, and feed into the various bedrooms. Another then goes around the house. Out back behind the basement, it goes through *another* splitter. One side of that continues along to the far end of the house, through a fairly corroded signal cleaner, and finally up into the living room. The other goes into the utility closet, where it goes through *another* splitter. One side of that is supposed to hit the cablemodem; the other hits *another* splitter/amplifier, which splits it three ways to feed the three main rooms in the basement.
Jesus. It's a wonder that there is any signal at all, by the time it goes through all of that. The playroom has been through at least four, maybe five splits, so it gets a weak signal only if the basement amplifier is plugged in. The living room is only going through two or three splits, but it doesn't have the signal amplifier, so it only gets snow.
I have to assume that this system was at least marginally functional at one time, but I'm genuinely surprised. I suspect I'm going to have to work with the cable guy tomorrow to puzzle out which splits go exactly where, and pare it down to just the few rooms we actually give a damn about. (Really, having cable TV in the laundry room isn't high on our priority list...)