The problem was that the main batteries (Ni-Cads) had run down so far that
there wasn't enough power to respond to the On button, then the backup died
too. After recharging the NiCads I still had to press the reset button
underneath and restore my data from (not too old) backups.
I still don't really understand why even with fresh fully charged batteries
it wouldn't come on until I factory-reset it.
Dave