Around 4 o'clock the association manager came and looked at my situation. He's pretty sure that the water's not coming from a busted main, given where the mains are located (he checked the plans for our building) and where it looks like the water's coming from. Thank goodness. They would have had to tear things apart in here to get at it.
He then checked the little meter-reader thingie attached to my pipes (don't you love when I get all technical?), and it wasn't reading like there was any water running. To test it, I went and turned on the kitchen tap. Nope, there's no water running unless it's been requested. So it's not coming from any of my plumbing.
Then he wondered if it was coming from my water heater, because it was feeling a little wet on the floor around its base. He asked me for a clean, dry, old towel--what he was about to do was going to pretty much destroy it. I found a tea towel that I don't care if I never see again, and gave it to him to dry the floor around the water heater. If water started showing up again in a few minutes, that's where the problem is. Boy, howdy, did that thing come back filthy.
A few minutes later: no water around the water heater. So he thinks that the problem is next door. He thinks my neighbor's water heater is leaking, and that the slope we've been built on has the water running into my unit, using the drain in my closet. Unfortunately, my neighbors weren't home to confirm this, but he's left them a note. I hope they come home soon.
I've asked him to keep me posted, so that I know when I can stop worrying the problem is on my end. He said he would. Though I suppose if I don't hear from him again, that'll be an answer too. Of sorts.
And now back to straightening up the downstairs. Though I have a bunch of junk in the living room that used to be in the closet. I'm not going to put it back in there until I know that I won't have to drag it out again. And besides, it's wet in there.