God, I can't believe how simple this is:

1. Sleep your computer in the middle of a backup.

2. Power it back on connected to another network (or no network at all).

This'll give you a "volume cannot be found" error. I ended up deleting my sparsefile, which produced an even more fantastic symptom where the computer will allow you to choose the disk as your TM backup disk, but will simultaneously slide the TM "power switch" to the off position.

I can't really say for sure how I ended up fixing it, but I went back to the machine to which the TM drive is connected, deleted and re-added my username for network permissions, then deleted my TM plist from the target machine. All is well now, but the fact remains: overall, TM is way too fragile for a consumer-oriented backup strategy.