Some folks have solved their FaceTime issues by switching their DNS service to either OpenDNS or Google Public DNS. It is best to configure it directly in your modem or router if you have that level of access. But often just configuring it on the Mac will override the modem/router settings.
Google Public DNS
I did see mention of using Google Public DNS while I was searching the forums prior to posting, but to be honest didn't think it was a DNS-related problem.
However, to rule it out I've set the prefered DNS servers on my router to Google's, but FaceTime still disconnects after 4-5 seconds when calling between me and my brother-in-law. It doesn't matter who initiates the call.
As a test, before I made the changes to my router, I logged into FaceTime as him on my MBP and tried calling myself on my iMac, and that was successful.
The same test between the two iMacs when they were here was unsuccessful, so now I'm definitely thinking it's something on the '08 iMac.
Now I just need to figure out what...