Trying to explain as simple as possible...
There are usually 4 thin fluorescent tubes in the monitor, two at the top, two at the bottom. These tubes work at around 700v but low current. Some monitors run the four lamps in parallel, others pair them, so you basically have two lamps in series and push close to 1500v through them to start them.
So the controllers basically work with two pairs of lamps and they have a sort of timeout period... they send power through the pair of lamps and they monitor to see if the lamps actually use current. If the current doesn't stabilize within a few seconds, the controller shuts down for safety reasons. If a wire at the end of one of those tubes is desoldered (and holding on just by the silicone jacket and sporadically making contact with the lamp metal pin at the end) the controller may detect there's no stable load on that pair of lamps and the controller will turn off after a few seconds.
Also, those tubes get weak as time goes by... but it usually takes years to become so bad as to cause problems. When a tube has too many hours, at the end of the tubes they usually become dark yellow or black and when they become this way, the current consumption of the tube increases. The controller detects this current increase and will also shut down, trying to protect those small transformers from overheating.
Now the controller needs stable voltage to power the inverter and the lamps... it gets this power from the secondary side of the power supply. If the esr of some capacitors there goes high, it's possible that when the controller pushes power through the inverter to the lamps then the bad capacitors may cause the voltage to go below a threshold and reset the controller or fool it into thinking the lamps are bad.
I would first check the end of the cfl lamps and make sure the wires are properly soldered - you have to be careful because it's hard to pull the silicon jackets from the end of the cfl tubes without breaking the wires and it's also somewhat difficult to put the panel back together.