Thanks Ralph and Matthew. This doesn't appear to be applicable to me. My unresponsive 5330e was built in week 52 of 2012, and the other phone is a 5360 (which is not affected by this issue).
On a related note, our tech just got back to me today and asked me to plug these phones into a known good network jack and plug a working phone into the jacks these two phones were on. If you'll notice, this is what I already tried, and not surprisingly I got the same results today that I got on Monday: The issue follows the phones. On the bright side, its nice to know that he is only a few days behind me on his efforts to troubleshoot the issue.