Because it was happening on two different devices (on my home wireless network), I decided it might be the DSL router. It's a Motorola NVG510 (supplied by the AT&T Uverse service).
I figured maybe there would be errors in the router log that would correspond to the times when the error occurred. But...nope. No log entries at all at those times.
Then, in poking around on the settings for the router, I discovered the "Advanced Firewall" settings. This is what was set on that page:
Because I didn't want to spend a whole lot of time futzing with different settings, I changed everything to OFF, like this:
Voila! This fixed the problem: no more errors of that type!
I'm speculating that it was a flooding issue, and that if I tweaked these settings a bit by adjusting the Flood Limit and Flood Burst limit, the problem could be fixed while still leaving in some of the protection that the Advanced Firewall provides.
But for now, I'm just leaving the damn thing off!