A careful search of the internet found a mailing list thread in which many, many other people had the same problem, all starting after 2006-05-12.
The thread starts here: http://firstname.lastname@example.org/msg09812.html
What turned out to be the problem? All these systems failed at the same time, exactly one billion seconds before the 32-bit Unix epoch ends in 2038. The timeouts set for database threads caused the software to look ahead, gasp in horror and died.
Ladies and gentlemen I'm in a select club of the first victims of the Year 2038 Bug.
My job is weird.