What is above the bathroom?

The attic.

First thing i would do is crawl up into the attic and see if you have a moisture problem, IE roof leak. If the ceiling above the spots is dry, then it is probably mold from poor bathroom ventilation.

Thanks for the advice. I will have to try to determine what the underlying problem is. In the meantime, what would be the best way to clean something like this. I realize it may not be the solution, but if I could try to control it until our remodel I'd like to.

There are specific paints for covering those spots. Kiltz is one. They keep the spot from reappearing. OTherwise the brown or mold will just eat through and reappear.
We have used Kiltz and it did the job. There is a better brand but the name escapes me right now.
I agree though you need to find the source.

simply mix one part bleach to 2 parts water and put in a spray bottle. Mist (not soak) the area. Repeat as needed. If it is mold.mildew, the bleach will work very very quickly (in minutes). You might wipe first with a warm, wet sponge first assuming the ceiling is painted with an acrylic (not flat) paint. Otherwise skip this step and just mist from about 6 inches away.

Thanks for the idea! I will try that as soon as I have some time :-)

I'd suggest that you also try the paint forum, since I've had spots like these in our bathrooms over the years (don't know if they were sticky, never touched them) and they were something given off by the paint, as I recall.

Of course, the ones that we've had have were removable simply by wiping with a wet sponge, so I don't know if these are the same thing that we had or not. We were able to just wipe them off, no problem. More appeared, but in different spots, and they were just able to be wiped off as well.

They've not been a problem, if they're what we've experienced.

-Roger