If you have single pane windows, it also could be caused from condensation forming on the inside of glass and dripping down during the cold months. This definitely happens at my house in Winter, it's a constant battle with mold and flaking paint and plaster until I break down and buy double pane windows. I tend to do a little plaster repair and repaint in the Summer and do cheapo shrink plastic around the bad windows, which helps a lot.

Yeah, and what I'm imagining with this flaking plaster business is just what Cliff is describing: the water is coming through the wall, not condensation. any water dripping off the glass would be 'harmlessly' drizzling on the outside of the paint.

So, why is water coming in?
Anywhere on the outside of your house where there is trim (around windows in this case) there is potential for water to enter the walls. Homes on the east coast are usually flashed correctly, but you need to take a look at the outside of this window. Bay windows have lots of trim so there will be lots of places where the flashing could fail. What is happening is the rain (or melting ice and snow) is going behind your trim, possibly following the casing nails right through the sheathing. Spend some time investigating it yourself, but hire a pro to do the work imo, this is a once-every-50-years repair: it should be done right. Speaking of which, and not to take this way way to far without even having seen it, I'm guessing the whole building needs work, and "piecemealing" this repair isn't really the right approach; the whole place should be examined and fixed at once. which brings me to:

You might consider calling your insurance agent and talk about filing a claim for water damage. I certainly would. And I have: my condo at Alpental, WA is being literally torn apart because of damage from snow. The repairs to two buildings are going to total over a million bucks! The stucco around my fireplace was doing that flaking too. I'd post a link to the photos of this project but I dont think they're super relevant.

Oh, and this just hit me: the British call Band-Aids "plasters" (i think) and Americans use "Band-Aid" as a metaphorical term for a temporary or makeshift solution to a problem, created ad hoc (often used with a negative connotation of a lack of foresight.. thanks wiktionary and thanks to all who spent the time to read this rambling post.