Water was flowing around the ceiling fixture and the fire sprinkler! (The shower curtain in the background is askew from Lisa using it to catch water and direct it into the bathtub instead of the bathroom floor.)
We pressed the room trash cans into service to catch the drips. Note that I took this video a little bit later, after we'd reported the trouble.
After discovering that the room phone wasn't working, I ran (literally) down to the office to tell the manager. He came quickly with me, and then headed upstairs to the room immediately over us. A few minutes later, the leaking slowed, then stopped. Our first theory of an overflowing bathtub was wrong; the tenant there was just taking an ordinary shower. Apparently there was a leak in the drains that poured down from our ceiling.
The hotel manager (a very nice fellow, incidentally) had the person upstairs move rooms and brought a mop and started to clean up our room. He also brought us a complete new set of towels to replace those we'd used to try and mop up the mess. As we're moving out tomorrow and have thoroughly moved in to this room, he said we didn't have to move, which was handy. We went and had dinner while he cleaned up the room.
I was a little worried that he'd think we caused this, and had visions of the famous hotel flood at Disclave that led to the little "don't hang things from the water sprinklers" signs in many hotels. But the manager saw immediately that this had nothing to do with us, and was just bad luck. He thanked us for reporting the problem so quickly.
It was a good thing that we hadn't gone to dinner first, as the mess of water that would have flooded our bathroom would have been much worse.