I have since brought the computer into my office, where I was
instantly able to hop on to an open access point that seems to be in
the neighborhood. This suggests that the problem must be with my home
router/AP. I don't know why this would be, as it's been running fine
for years, has the most basic configuration, and is working with
several other computers. But I'm not sure what else to think about.
Let me guess: You're accidentally connecting to a neighbour's open AP
who's using the same 192.168 addresses as you are. Your neighbour has
dialup, or unplugs his DSL, so you can't really use his AP.