A bad or intermittent connection can cause through repeated starts the compact fluorescent to have a reduced life, with PWM LED lamps it could affect the regulator chip however in neither case would that explain why the fuse ruptured. In theroy every bulb should have a built in fuse or at least those with BA22d fittings as in the main the bulb holder only rated at 2A and fuse rated at 5A but I know from bitter experience this is not always the case.
So I would first look at the bulb. I know I had a problem with bulbs from Ikea it would be nice to say don't use cheap bulbs but again experience has shown expensive can be as bad as cheap. So I would swap supplier to start with and see if there is an improvement. Even if the bulb fails prematurely because of a bad connection it should not cause them to go short circuit. If it is a bad connection causing bulbs to fail I would think the LED type would be less susceptible to damage although can't be 100% sure on this, but I would consider trying an LED bulb.
If you can find a tungsten bulb to fit the holder this may help identify the fault. The extra load would likely cause a bad connection to get a lot worse but not sure if this would be good or bad as bad connections can cause fire under heavy load. Problem is we don't really know if bulb failure is due to bad connection or simply poor quality bulbs.
i stand corrected! i am sure (it was a while ago!) that we did an experiment in college where the motor was pulling more current when we simulated a less than perfect connection but will bow down to the knowledge of the forum
A motor can draw more current when it has under voltage or loses a phase. The same could be true with pulse width modulated or switch mode power supplies as they compensate for the volt drop so yes a LED bulb say 5W would draw 22 mA at 230 volt but 33 mA at 150 volt. Under 150 volt it will likely simply fail to run and even if you had a 50W LED lamp it would still only draw 330 mA at 150 volt which would not rupture a fuse.
So in real terms bad connections should not cause enough extra current to flow to cause the situation described. With a standard wire wound ballast on a fluorescent again it would not cause any extra current to flow. But with a HF ballast then the spikes in the voltage due to a bad connection could cause electronic components to fail. So it is possible spikes due to bad connection could cause diodes in the HF ballast to fail which could fail short circuit. I personally think it is unlikely but will accept it is possible. In the main I think the CFL uses an electronic ballast so could be affected by spikes.
I find a standard radio is a good tool for finding if there is a bad connection as spark transmitters will cause interference better to use Medium Wave if you get interference when lights are switched on then either bulb or wiring is faulty.