Newly installed ceiling fan, lights turn on and off on it's own - dimmer switch, breaker or light bulbs causing issue?
If the breaker isn't tripping (all your bedrooms are not losing power, just the light on the fan), it's not the breaker being overloaded.

How is this wired? Is the "dimmer switch" controlling just the lights on the fan, or is it the older fan's speed control? Or perhaps if I ask "How do {or did, with the old fan} you turn on the lights alone, fan alone, or both?" - is there a wall switch or speed control for the fan, and a separate wall switch for the lights, or does one wall switch (dimmer?) turn on the fan/light and then you select if the lights are on by a pull-switch on the fan? You have a remote control now - that would probably affect the way things should be wired.

Why do I ask all this? The only way I can imagine that you're causing something to overload with 120 watts of light bulb is is if you've got the fan motor attached to the dimmer, which is almost certainly not right...either the "dimmer" is a fan speed control, probably sized for the previous fan motor (did it have lights?) or it's only supposed to be attached to the lights, if it is a dimmer. 

A dimmer won't work right as a fan speed control, and a fan speed control won't work right as a dimmer, and if both the motor and lights are attached to either one, it would not work.