Short version: there was a small error, which became a large error because my tool to detect errors didn't work right.
Longer version: there was a need for our provider to restart the SAN. The watchdog reported a 2 minute outage, and then full service restoration. Unfortunately, it seems our network configuration was damaged—the watchdog could see it, as could some users (including me), but most users probably could not.
I opened a ticket with the provider as soon as I found this out, and we have a temporary network configuration that should admit most users. There may be intermittent issues as they work out the full solution.