Some details will help. Is it raining? Has it rained? Are the gutters clogged? Is there a bathroom or kitchen above the area the water damage is located? Has any work been done to the exterior of the house in that area?
Try these for a start.
Ron

There's no kitchen or bath above the area. No work has been done to the exterior. There's a dormer above the family room area where the window is, but there's no water on the floor. We had a little bit of snow, but nothing to speak of.

Sounds like a humidity problem to me. Above a window
is what is called a 'header', which is framing lumber,
behind the sheet rock, leaving little room for insulation,
thereby making it cooler so humid air will condense on
it.

It sounds counterproductive to patch the damage without establishing the reason. Usually such problems will reappear. While not fun, might be best to cut out the drywall in order to see if you can detect the water source. Roof leaks sometimes travel remarkable distances before they manifest themselves. You also want to be sure you don't have wet wood that can lead to mold or insect infestation.

I would bet it is a roof leak. Could be a plumbing vent pipe boot up on the roof, or could be flashing or a valley around the dormer, or it could be many other roof related leak areas. As said, water can travel far and wide from a roof leak. It can bypass the second floor altogether and dump into the first floor. I'd go up to the attic if possible and search around with a flashlight. Look for water stained wood, wet insulation, or cracks in the plumbing vent pipe boots.

could be that little bit of snow, especially if on north side of house, couldn't see it from the outside, but I had ice back up under the shingles and the melted water was running inside the kitchen cabinets, with no apparent problem on the second floor.

This gets a little strange: I noticed last night some of the paint peeled off of the ceiling, and I felt water. I hammered a nail into the wet area; water started dripping out and continued for about an hour. This morning, there was no more water dripping, and no further bubbling on the ceiling.

Have you gone up to the attic to see what is visible from there, yet? Is the attic accessible? Did it rain last night, or snow? Are your gutters clear? How much water came out from the hole you made with the nail - a tablespoon, quart, gallon?

All of this is useful to know. My instincts continue to tell me this is a roof leak of some sort. Even if it hasn't rained in a few days or more, water could have been standing there in the ceiling, softening the sheet rock until that paint peeled.

Now that you have a clear nail hole in the ceiling, watch it next time it rains. In fact you may as well enlarge the hole by using a keyhole saw and look around up there (you're certainly going to have to cut out the damaged sheet rock anyway for a repair, after the source is found). If it is a roof leak or the gutters backing up then I'd guess you'll know it if water comes down next rain (or melting snow) event. By the same token, I encourage you to go into the attic during the next rain event to see if you can spot a leak.

You may as well explore the header area of the window, too, as was suggested to see if there is excess humidity. I'd bet if that is true there'd be a colony of black mildew up there ready to greet you when you expose the header.

If you are certain that the roof isn't leaking and the header is clean, then most likely this is a waste water pipe or drain problem originating from an upstairs bath. Water can take unexpected downhill pathways. Doubt it is a water supply line leak or it would probably be a constant leak and you'd have a royal mess by now.

We checked the attic, and moved stuff around, but saw nothing.

It rained last night, but after the water stopped dripping.

Checking the window header is a good idea; we'll do that too.

I doubt also that it's a water supply leak, because it would have gushed all over the place.

My gutters are on the front and back of the house; the window in question is on the side of the house. Does that make a difference?

"My gutters are on the front and back of the house; the window in question is on the side of the house. Does that make a difference?"

Yes, that would very likely rule out any difficulty from gutters backing up since the gutters are not on the side of the house, where the window is.

Checking the window is a good idea since there is evidence on the wall of water infiltration but you're still left with the difficulty of locating the source. If it were only the wall that had water spotting one could predict it is perhaps a difficlty either with moisture build up or faulty caulking around the window. But since you have a spot on the ceiling seven feet away from the window, that means that fixing faulty caulking probably isn't the total solution. However, water can travel across a ceiling from an outside wall if there were even a very slight slope toward the center of the room.

You went to the attic and saw no problem. Was that before the rain came or did you go up during the rain? Were you able to put your eyes on the area around that dormer window that is above the living room? Roof leaks can be challenging to spot if it isn't actually raining (or snow is melting on the roof) at the time you go up to look. Too, if it just a sprinkle of rain outside you may see nothing. Even if it is a downpour of rain, if the wind is blowing the "wrong" way, the roof may not leak at that time. That contributes to the frustration of finding the leak.

I had a leak once that only presented in downpours accompanied by a strong NE wind. Same rain with a SW or even SE wind caused no leak in that case. It turned out to be chimney flashing, on the NE side of the house. And it did no damage to the second floor of the house. Only showed up at the base of the fireplace.

As has been said, it may turn out not to be the roof at all. Could be the house's siding leaking somewhere above the window. Even brick siding can leak at cracked mortar joints when the wind blows the rain that direction.

Keep after it and keep posting - someone is bound to help figure this out!

A surprising amount of water can be produced by condensation over time. We have had a couple of situations where there was water from condensation, and the source of it had to be remedied before the problems it caused can be permanently fixed.

If you don't see any leaks or signs of them, that would be what I'd be looking for. There are experts in this area that can look the situation over (for a price), and very often can find problems by leveraging their wealth of experience. Last time this happened to us we hired an expert, and they found the condensation problem after a lot of work. It has not recurred.

-Roger

Sounds like what I've had. A contractor told me to turn down the humidifier on the furnace, but I think it came back. I went back to the attic and crawled waaaay back into the perimeter and hit the area with a light and found some water stains-where there was no insulation. I rented a blower and attacked every nook of around the edges with more insulation and it helped.

However just yesterday I found three of the splothces were wet once again. Maybe I didn't get the insulation back far enough, but I had turned the humidifier back up a bit too. I think the insulation may be my problem since it only happened yesterday in three of the locations rather than all of them.

I've lived here for almost three years and last summer I added roof ventilation (ridge vent and undereave vents) and then my condensation problems kicked in. The fact that the dampness ended in many of the areas tells me that I'm on the right path, but man this is getting annoying.

I spoke to the company who installed our new roof 2 years ago. They're coming tomorrow to take a look. The guy told me they're very good in finding leaks. Keeping my fingers crossed, and will keep you all posted.

Thanks for all the advice.

I also called my plumber, who's coming tomorrow. My husband was away all week; when he came back last night, we both checked the radiator in the dormer, which is over the family room. We felt some moisture on the pipe. We hope between the roofer and the plumber, we can get to the bottom of this.

P.S. - The ceiling dripped last night (after my husband turned the thermostat up) and this morning (after he took a shower).

This is a real mystery that I hope has a easy solution.

The plumber came on Saturday. He took one look at the radiator in the upstairs dormer and said, "your bleeder valve is broken". He replaced it, which took all of 5 minutes.

No more leaks.