Thanks for posting!
We don't normally recommend this configuration but, as a test, you may want to try installing Pro on the server machine and opening one of the databases (still through Open -> Remote). This would help rule out the network as a variable.
I may have solved the issue. I did several things, but I'm honestly not sure which really fixed it.
*I reformatted the computer and installed all the software again from scratch (OS X 10.5.8, Filemaker Server 10.0.3, and iDrive for Mac). When I installed the Java Update 9 for OS X 10.5, the Filemaker Server Console wouldn't launch any more, but I did a archive and install, reinstalled all the updates minus that one and all is well. - I'd recommend avoiding the Java Update 9.
*I set the log out after idle time to 12 hours, which had been set to 1 hour. Previously, it wasn't set and they were still getting kicked out, so I'm not sure this actually has an effect at all.
*We also had Comcast come replace their modem, they apparently weren't able to log into it remotely.
*Finally, our server uses an public static IP address and someone had entered the wrong network settings for the gateway and DNS servers. When I entered all the correct info, the server worked and I haven't heard a complaint about people being dropped since.
If you're having this issue and have the time (I didn't), you could try each one separately to find which one is actual the fix.