It could be the ballast. Apparently they don't just go kaput one day, they keep get weaker.... Could it be the ballasts going bad in the fluorescent lights? We have one down stairs in a finished basement room and it would do the same thing. Now when it comes on it is a very very dim light. I have changed the fluorescent bulb (one of those "U" shaped bulbs) and still get the same thing. So I am guessing the ballast has gone out. After all, it's over 35 years old!
I spent a lot of time searching last night. Looks like a lot of people are having trouble with humidity causing corrosion of the metal on fluorescent bulbs and sockets. I tried new bulbs, and steel wool on the prongs once, but it didn't last long. They make corrosion resistant tubes, probably very expensive and special order. And you can buy LED bulbs that retrofit, but my light fixtures are very old also, so their days may be numbered anyway.
It looks like the best bet might be to replace with LED fixtures. Several people mentioned they keep changing them, and older versions become outdated, like phones and computers.
Here's some strange advice I found. It actually worked for two people on a DIY forum:
"I've seen the humidity thing. It's probably a cheap/bad magnetic ballast that can't put out enough voltage to establish an arc except under ideal conditions. Brand new fixtures can exhibit this problem. I'd recommend a new electronic ballast, however there is a trick to making a weak ballast work; Tape a thin strip of foil along the length of the back side of the bulbs. Do not allow the foil to touch the ends of the bulb. The foil and the glass tube form a capacitor to help establish the arc."
I don't understand it, but think I'll try it just for fun.