After reading this and similar threads on other forums, I feel like it's pretty safe to conclude that the problem is the HR10-250 itself. Something is just plain sick in its innerds.
Why?
Happens with HDMI
Happens with component
Happens with S-Video
Happens with various TVs/monitors
Happens with various dishes
Happens with various multi-switches
Happens when watching live TV
Happens when watching recorded programs
Happens when not watching anything, just navigating Tivo menus
Doesn't seem to happen when:
People use a different HD receiver
People use an OTA and a different box
And since they've phased this box out, they don't care, they never have, and there won't be a fix. All we can do is hope the HR20 is better (at least in terms of dropouts - we know it won't be better in terms of DVR interface...no Tivo!).
For what it's worth, I hooked up my whole system to a UPS and the problem diminished significantly. For now....
My $0.02, y'all.
