I've tried various voodoo solutions, including rebooting one or both computers. Some times things get better, some times they don't.
I think I may have figured out the problem. I have my networked laser printer hooked into the KVM switch through a USB port so that I can access it from the work machine since that box can't see the home network at all. I tried to print something from my home machine and discovered that the printer wasn't responding, nor had anything that I'd printed in the last few days printed out either. This usually means that it's time to reset the printer by turning it off and back on.
A bunch of jobs from various machines on the network started cascading out of the printer. And when I went to switch between home and work computer the next time, the mouse responded immediately.
I suspect that the problem is that the USB support on each computer is desperately trying to ping the printer. The printer is jammed and not responding. This stops the USB support on the computer from working on down through the devices and finding the mouse until the attempt to talk to the printer ("Bueller? Bueller?") finally times out.
We'll see if this theory holds...