Update.
We may have found what caused the issue, but dang if I understand why. We installed some new copiers back when this problem started and like all things that break, the first thing you look at is what changed recently. We unplugged 2 of the copiers and this had no effect on the problem, and they were a different subnet than the phones anyway, but on the same physical network and same switches. There was a third copier that had just been moved, no changes made to it. I just left it alone and like we reported earlier, the phone problems eventually just went away.
However, we had other strange network behavior, nothing critical, where a connection would drop for a brief instance and then come right back, but the phone system was not affected. Finally, we discovered that the third copier that was moved retained its IP address, and one of the new copiers that was installed in its old location was setup with the same IP address. I had given the copier guys a different address to use, but they just printed up the config from the third printer and used that one instead. When we reset the IP address correctly, the intermittent network connections stopped and just for fun, I set the copier back to the same IP address and plugged it back into the network and as soon as I did...phones started the Network Connection issue. I quickly unplugged the copier and no more problems since. I don't understand why a copier with an IP address of 192.168.100.223 could affect the phone system that was using 192.168.101.x, and why it would stop affecting it after a random period of time...but that seems to be the case. Live and learn. Be careful with those static IP addresses.
