I assume the lights are in pairs then? Does it happen with both sets? Waterproof endcaps on the fixture? Have you tried new bulbs?
Most newer T5 light setups have ballasts with EOL (end of life) protection. This means it will no longer attempt to light the bulbs if they are worn out. I have a Hagen Glo 36" 2xT5 fixture and it will do similar to what you are experiencing when one of the bulbs is gone. It will light for a second or two and then turn off. My fixture doesn't have any switches to play around with though, it's just on a timer.
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