I called someone who knew more about these things than I did, and he checked the same thing. We remoted into the machine, and it has two network cards in it, with two IP addresses. One IP address returned pings, and somewhere along the way, the other one started returning pings, too.
One IP was dynamic, the other static. "What building are the servers located in?" the network guy asked. I told him. "That IP address doesn't sound like it belongs in that building." I checked with my team lead -- yup, that's the right building. "How long has it had this IP address?" I checked the properties of each. Four days plus. "Maybe disabling and re-enabling the dynamic IP network card will get it working again."
I clicked the disable button. Nothing appeared to happen. Clicked it again with no response, not even a button press.
It was at this point I realized I'd been sawing on the branch I was sitting on. In a cartoon, this would be the moment just before I realize I'd walked off of a cliff, but gravity hadn't taken effect yet.
The remote connection closed. Gravity took over. "Connection failed."
Road trip to the server building. Checked in -- and the server wasn't there. Wrong building. That's why the IP address didn't sound right.
Road trip to the other server building. Wait 20 minutes to be checked in. Find the server, click to enable. Remote access reactivated.
Note to self: don't do that again.