Yes, if the interior ceiling is getting wet at the wall and radiating outward this may be the cause. This is a common occurrence in high wind areas especially with the use of vinyl or aluminum soffits or soffit systems that are not inset from the fascia system.
One solution is to replace the soffit with solid wood or solid fiber cement, and provide ventilation through a continuous vent that is only a few inches wide and centred towards the wall. Or install individual vents that have louvres, make sure the louvres are pointing towards the wall. Another solution that may help is to use a gutter there that provides adequate drainage.
The solution really is dependent on several factors, the height of the soffit relative to other parts of the structure, or other roofs around or underneath the area. The exposure and strength of the wind, and the size and design of the overhang also should be considered.
I would venture a guess that @Tester101's suspicion about your floaties coming from your glass (unrinsed soap, dust, etc.) may be heading down the right path. I would try rinsing a glass, scrubbing it with a new sponge or clean washcloth, wiping it dry with a clean towel (not paper towel, as this may leave pieces of paper behind), and then refilling it. @NiallC's comment about asking for a water report from your municipal water supply is a pretty good alternative, but doesn't factor in anything that may come from the pipes between them and you. Most health departments will do water testing, so you could always fill a bottle and take it in.
Beyond that, if you really have to know what's in the water, your only real solution to absolutely know with 100% certainty (technically, unless you watch the health department do the tests, and can verify their results, you can't KNOW that they're right...but I'd probably just accept it) is to get testing kits for this, that, and the other thing (that was just the first page offering a wide range of kits that I found), that operate similar to what you might use for pool water. Depending on the sensitivity levels, you may pick up on things like trace amounts of chlorine...but of course this is going to get back into the more expensive end of things, as those kits aren't cheap (Grainger's seem to run around $40-$50 and up...it's about $20 for their cheapest, which is testing for copper, on that first page of results).
You could pick up a pool test kit, which will cover a couple of parameters (chlorine, calcium, hardness, etc.), but it's not going to cover everything and may not operate in the range you need to identify your floaties...it's designed primarily to help you maintain clear-looking pool water that you're not drinking, not crystal clear tap water.
So really:
Ask for report from municipal water supply
Take in sample to health department
If you're still paranoid and want to know what it is, buy test kits
If you just want to get rid of the floaties, filter, filter, filter, and make sure your dishes aren't the source
Best Answer
What you're looking for is called "water contact indicator tape". It's used in cellphones and other electronics to detect when damage was caused by water for warranty-rejection purposes. You get it wet, and it turns bright red, as shown on page 2, and stays that way permanently.
Though it's generally only sold by the 100-count reel.