Here's what we have done to isolate the source of the problem, although I'm not sure that we have corrected it. I had a colleage ping the ISP's default gateway (the one that is handed out by DHCP to the Linksys router) from the outside while I pinged it from the inside. The problem is definitely between Comcast's default gateway and the client machines.

I swapped the two 8-port switches for new ones as it appears that the problem is on the internal network. It will be next week before I can do more troubleshooting, but PingPlotter has been a great tool for this effort. Thanks for the suggestions.