Why are my fluorescent lights not working sometimes?
Confirm your wall switch is good. Bad switches can be intermittent.
Turn off the switch and take the tubes back out and
put them back in making sure they're fully snapped in. It's worth a
try. Don't go buy more, but if you have other new tubes try them.
If the lights work for a period of time and go out on their own,
that usually indicates an over-heating ballast - replace the
ballast.
Otherwise, the best test for a ballast is just what you did, try new
tubes. If no luck, replace the ballast.
While replacing the ballast, make sure all of the wires in the
sockets are well connected. Give them each a little tug. If you can't get a
wire to stay in a socket replace the socket, maybe replace all 4.
Fluorescent lights need to be grounded to work properly. Make sure a
ground wire is attached to the fixture. I've seen fluorescent lights, especially in old homes, where the people say they're always replacing tubes, and it's because the fixture isn't grounded.
If you're going to replace the ballast or the entire fixture
consider going with an electronic ballast maybe even T8 size tubes.
Amazon sells the smaller clips you'd need for a retrofit to T8 U-Bent tubes - ASIN
B00A3WGR0W.