When I was halfway through the fourth filter (out of five), Emerson started banging on the bathroom door. It seems that Chuan-Kai had been awakened by the smoke alarm in his room, which was going off because there was water dripping from it. Oh, crap.
So I stopped washing the filters, abandoned my plans to shower and go to work this morning, and called a plumber.
Around 8:30, Ken the friendly plumber* arrived at my doorstep. He was pretty remarkable. In about a half hour, he had cut a 6" x 12" hole in Chuan-Kai's ceiling, exactly underneath the tub drain (I have no idea how he was able to pinpoint the location so precisely). He explained that the water usually runs along the floorboards until it finds the nearest opening, which in this case was the smoke alarm fitting.
Then he had me run water in the tub while he checked for leaks. Nothing. The pipes were bone dry and the wood in that area showed no signs of water damage.
Eventually we figured out the problem: when I'd put the rinsed filters on the bathroom floor, there were still dribbling, and the water had run along the tile floor to the wall, where it had dripped into the ceiling below. Solution: don't let inordinate amounts of water drip onto the bathroom floor. I can handle that.
Since he was around, I had him look at the downstairs bathroom sink (which started dripping last Friday). That took him a lot of effort to fix (because the sink has this terribly inconvenient decorative column hiding all the plumbing underneath it), but no surprises. By 10:30 it was all done.
*Ken is from Apollo Drain and Rooter Service, Inc., whom I wholeheartedly recommend.