Troubleshooting a String of LED Recessed Lighting Fixtures
I think what everyone is suggesting is that there is something else wrong other than your cabling. If I was called to your home and you described this problem, I would not just come out and say "we need to replace the cabling". What a good service electrician would do is try and isolate the problem.

He would probably pull the switch and inspect it and put a meter on it just to make sure it was working properly. Then he would go about disconnecting all of the recessed cans inspecting each one for damage or burnouts. Once he finished disconnecting the recessed cans. He would then test all of the wiring for any problems. If it is not the wiring or the switch (which it probably isn't) and he hasn't found any defective splices or damage to the can lights, then he would begin to connect each light back up and see if he can duplicate the problem.

What a really good service electrician would do first is to call the manufacturers tech support and first find out if this has occured before and get their input.

Good luck