Thanks for the link. I haven't had a chance to do much more troubleshooting yet, but I was able to get my admin interface up. When I did open that, I got a series of these messages:
I'm not yet sure if it is EVERY device, or just those in this part of the house. I have to do a little more experimentation. For what it's worth, I have an ISY99i and a 2412S PLM. These are both at LEAST 6 years old (how long I've owned this house). The PLM does have power, at least the little light is on. And, like I said in the original post, if I click on Fast On or Off like 50-60 times, I can still get a device to react.
I can't think of any changes - things plugged in, etc. - that would have led to this, but I'll walk around and look at every single outlet and see what I find.