If the spots wipe away easily they could be rust-related, but also could be condensation-related. The nail or screw heads are a direct connection to the rafters or joists above, and therefore conduct more heat/cold. Humid air from bathroom activities hits the spot over the screw/nail head and water forms, often collecting dirt from the air and surface to color it brown. Left long enough the spot will become a drip if this is the case. Either way you could have a moisture and/or ventilation issue. A timer bath fan is a good way to help with ventilation.
You mention no smokers in the last decade, but nicotine and other side-effect of cigarette smoke can really get into surfaces. If you want to try an easy solution, try coating the entire ceiling with an expensive stain-blocking primer, and fresh paint, gloss or semi-gloss. This may seal in the smoke residue of that's the case.