@srlebert,
It seems like there might be, from the description, a heat buildup problem that might be causing a continuity problem.
If it were me, I'd pull them out and either re-wire or closely examine the fixtures and their wiring.
Being familiar with a voltmeter as I am, I'd check to see where are the breaks in continuity.
Then if I couldn't spot the problem, I'd hire an electrician.