Lighting – causing a two brand new fluorescent light to pulse

1. Confirm your wall switch is good. Bad switches can be intermittent.
2. 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.
3. 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.
4. Otherwise, the best test for a ballast is just what you did, try new tubes. If no luck, replace the ballast.
5. 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.
6. 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.
7. 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.

Fluorescent fixtures can be rather tricky. With that many installed on the same circuit, what you might be seeing is an unbalanced load on the circuit, causing some of the fixtures to have full current and others, less, and quite simply a ballast, or any fluorescent will not work without full current. With the warm up times of older ballasts, combined with different features of newer electronic ballasts, this could be the answer.

Isolating part of the fixtures on the circuit from each other quite possibly will solve this. The simplest way would be to install two (or more) switches instead of one, preferably running from different circuits.

You might also consider replacing some, or all, of the can lights with LED. This would considerably lower the load, as well as saving energy. Quite interesting that you have can lights with an external ballast, as well. I've never seen that. What type of bulb do these use if not a CFL? Is it possible that what you think is a ballast is simply a wiring box where the can lights connect to the wiring, or some sort of dimmer control?