Ralph, after your initial recommendation (and before I saw your follow-up), I contacted Apple support.
I opted for the Chat version of Apple Support, started a session, and after 35 minutes of a repetition of "You should try this” and I’d say "I did that” (yes, very script-based) — he came to the conclusion that the only solution was for me to wipe my computer and start with a fresh install.
I was having none of that. I had no doubt that the problem was at Apple's end, and didn't want to waste the considerable time a "nuke and pave” (followed by a prolonged period of getting things back to the way I wan them) would entail.
I told him that the evidence is that there’s a problem with some servers at Apple, to which he replied that it wasn’t possible because there are so many users and (paraphrasing here) if there was a problem at Apple's end, there would be a lot more support calls. So it had to be a problem with *my* computer. Convinced that this wasn’t going to get me anywhere, I asked if I could continue the discussion at a later time, he told me that I could reference the ticket number when I resumed with them.
Later that night, I tested it again (and it failed again), and so I tried a few more tweaks to my computer -- like permission repair (really not necessary because I have it running daily, automated under cron control), did a Safe Boot followed by using Titanium Software’s Maintenance application to clear all sorts of caches (which required yet another reboot). I had no expectation that any of that would work, but I also wouldn’t want to skip something that *might* make a difference.
And to my utter amazement, after that last reboot, I was able to log into iMessage!
In light of that new “evidence,” I thought I would be able to solve the problem for my wife’s iMessage access too (by following the same sequence).
So I walked over to her iMac (which I hadn’t touched all day), and from her account (on her computer) tried to log into her iMessage account — just to establish a baseline (before I started making changes).
And I was able to log her into iMessage as well -- It "spontaneously" fixed itself!
So, I have to conclude that nothing I did to my own computer made any difference at all. And now reading that “five_eight_nine” can now access iMessages (with no apparent connection to any action taken there), my current hypothesis is that someone at Apple noticed that something in one of their servers wasn’t working right, and fixed it.
I only wish that the dashboard that Apple provides for such outages
http://www.apple.com/support/systemstatus/
showed a problem at their end (it’s always solid green), so I (and others) wouldn’t have wasted so much time troubleshooting a non-existent problem.