I've taken apart a CFL and they contain a rather crude switching power supply.
Three things or combination of things could be happening:
Capacitor is shunting excessive harmonics
Main switching transistor is overburdened
Main rectifier is overburdened with the harmonics (cheap ones cannot handle much more than 100Hz)
Switching PSUs will attempt to maintain output regardless of input...and dumb ones do it until the switcher fails due to high current! Such failures are often violent with the transistor actually exploding and leaving in some cases just some half melted leads on a scorched board!
The above poster is correct, filtering the harmonics will not work unless the power supply is designed to properly respond to a low supply voltage. It will still fail, it might take longer to fail, but I'd be surprised if it lasted longer than a day or so.
All of these potential failure modes could be prevented by simply adding a thermal fuse and then mounting it to the main switching transistor. Since the thermal fuse will also blow if subjected to excessive current, you can protect against the diode and cap failure mode too as they will typically fail shorted and pop the fuse.
