Trailer Light Question

Our 2010 Cougar or our 2009 RAM have begun something very perplexing. After approximately 10 minutes of use, the clearance/running lights on the trailer stop working. The brake lights work, the turn signals work, the brakes work, and the indicator on the brake controller confirms that we are hooked up and that the circuit is complete. I have tried disconnecting and reconnecting the lead and I have tried cleaning the connections. My first attempt at cleaning the connections appeared to work or at least that is what I assumed. I now think it was just busy work as it was done before I began to drive away. The lights going out has now occurred repeatedly in all weather conditions. When they are working, all of the lights are burning. Again to summarize...

I connect the lead and all the lights work.
After about ten minutes, the clearance lights (tail lights included) stop working.
The brakes still work.
The brake lights still work.
The turn signals still work.
The brake controller indicates all is well.

What the heck am I overlooking?

Signed
Currently Dimwitted