You have 120VAC feeding the transformer via 2 3-way switches.
You will need a voltmeter to diagnos the problem as well as access to the J-box where the 120V ties into the transformer. Check for 120VAC on the primary lead of the transformer, while operating both wall switches. If at anytime you loose power except when the switches are off, you have a problem with the switch, switch wiring or wiring to the transformer. If this checks out; check for 24VAC or VDC on the secondary side of the transformer when the wall switch is on. If no 24V, then the transformer is bad, although intermittent, you may have to play with the switches until the fixture does not come on and then start testing. If everything checks out from there, the only thing left is the wiring on the fixture to the lights, the lights themselves and/or the light sockets. Give these items a visual inspection and check for continuity and shorts from point A to point B for the lighting.
That pretty much covers all components involved with your fixture and switches.