The problem is that when the computer goes to sleep, it does not wake up properly. It starts to come out of sleep and then hangs, sending no signal to the monitor.
I've hunted on line and discovered that many people have had this problem with Windows Vista (and other Windows versions), but solutions appear to be mostly voodoo. One suggestion was related to video drivers -- I've updated them to the latest version, but that didn't help.
I will probably continue messing around with this at some point, but -- for now! -- I've just told the computer not to go into sleep mode.
In completely different news, it will be interesting to see how long it takes Katie and Julie to go into sleep mode tonight.
I predict that they will not hang in the morning when coming out of sleep mode.
Merry Christmas, everyone!