I have not yet found a solution to this problem because one thing I want to try, buying a new shorter cable, I can't do yet because the stores aren't open.
But here's something for those of us who can't get this thing working to try: use a shorter DVI cable, one that is not more than 3 m (9.8 ft) long.
In doing quick research, I noticed that the official maximum allowable length of a Thunderbolt cable is 3m (118 inches). DVI cables, however, can be MUCH longer. Official DVI specs say 5m (16 ft). I just measured my own DVI cable and it's 4 m (13.3 ft), so it is longer than is allowed for Thunderbolt specs. In reality, there are MUCH longer DVI cables in existance since many graphics card and monitor manufacturers produce devices that output much stronger signal.
Considering that the MacBook Air is a small device, I'm going to assume that it can only put out up-to-spec signals for Thunderbolt port. I would assume that means don't use a cable length longer than 3 m (9.8 ft). I'm off to Radio Shack this morning when it opens and see if I can get a shorter DVI cable to work with the MBA and monitor.