Some interesting thoughts, guys, thanks. Here's what we can see thus far:

The clients affected vary more or less randomly.

Putting a count in our SPID-killer sp is a...

No recent changes to client or server software. Certain clients do contact certain servers by default (basically, we split the country in two).

This did seem to occur...

Response is still the default of 20 minutes. We've considered lowering it to less than a minute, since all the normal queries the machine receives are quite small, but...