Sounds like power-supply regulation problem, could be a bunch of things in the 'chain'...

Could also just be a semi-corroded connector (VERY common) that suffers during higher current demands.
Might be as simple as the main filter cap gettin' old...

I'd suggest measuring the volts on the transformer (that eventually leads to the lamp supply) and make sure the transormer is rated well enough (not lowering in volts) as more lamps come on - design fault?

If volts are stable, then measure at the bridge rectifier, then the filter cap output and eventually into/thru the regulation transistors etc etc......

Somewhere along that line you will see an obvious voltage change as you bring on more lamps.

MM.