What's causing every light on a single breaker to dim and turn off randomly?
Ok I figured I would give this a shot. First I like Harpers suggestion. It's important to know if they are all doing this at the same time or randomly. Second what type of light bulb are you using? Incandescent, fluorescent, or LED. This is important to understand if the problem is consistent with the lamp. 

Finally, except for the dimming, you are describing what happens to recessed cans when the integral thermocouple begins to weaken in recessed cans. These are installed in all commercial cans to protect from overheating. If you disassemble the can trim you will be able to see it mounted on the side of the can assembly. The only way I know to repair it is to contact the original manufacture and replace the assembly.

Hope this helps, good luck.