I don't believe boiling water/broth or grease smoke would congregate and cause small rings like that in one specific area. It would hit the ceiling and spread causing a wide gradient of discoloration. BUT anything is possible.

Personally, based on what you just told me, this is what I would do if i were you. First try very lightly wiping the area with a dab of dish detergent in a slightly damp cloth to see if it collects on the cloth AND reduces the rings. Since the stove is there, i'm guessing you will pick up some grease regardless. You may have to do this again to confirm. This will help to rule out or confirm the grease possibility.

If that doesn't do it then monitor it. You can also go to a local hardware shop or amazon and pick up one of these guys (https://www.amazon.com/General-Tools-MMD4E-Moisture-Digital/dp/B00275F5O2/ref=sr_1_1?ie=UTF8&qid=1502224503&sr=8-1&keywords=general+moisture+meter). They're relatively cheap but are great to have around. Plus they do a much better job than feeling around. All that being said, monitor before you cook/boil. If you want to go to extremes, monitor right before and right after you shower.

If it isn't grease, and monitoring it over time shows no signs of large swings in moisture fluctuation than the water spot could just be old. Either a problem was realized and fixed or a one time thing happened and a previous owner never painted over it. Think 15 years ago if a kid overflowed the tub.

Good luck and update!