Title says it all about the problem, or almost. Started happening on nearly every reboot.
First fix was to open the ATI Catalyst Control Center. That auto-fixed the problem. Worked twice.
Next was to unplug the DisplayPort cable from the video card. Worked maybe three times.
CCC started saying the monitor was an Acer 0v23W, or something like that. EDID problem, likely.
Tried a registry hack that purported to add in unsupported custom resolutions via the DALNonStandardModesBCD1 registry key. Didn't work, my new resolutions never showed up anywhere.
Found some kind of editor for hacking ATI drivers. Didn't use it, it scared me.
After much poking around, I found the rather silly suggestion that unplugging the monitor would fix the EDID issue. (I lost the link, unfortunately.) The author suggested an unplugging time of 10 minutes would be necessary.
For lack of anything better to do for a minute, I gave it a go for 10 seconds. (Because the whole "unplug for x minutes thing to let the capacitors discharge" thing died somewhere in the 1980s, guys, that's why.)
And hey - it worked! Silly trick to fix a silly problem. Works for me.