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.
It's possible that the switch does power an outlet, but that the installer did not remove the fin that connects the top and bottom outlets. When the fin is removed, the top and bottom outlets are isolated from one another so that they can be independently powered. If the top and bottom outlets are wired with two wires of the same phase, you would not notice a problem with day to day use.
If you have a voltage tester, test to see if you have power to both the top and bottom terminals of the switch when the switch is in the off position. If you do, it's likely the installer just forgot to take a fin off one or more of the outlets.
There is probably a way to test for this without any tools, but I am stuck at the moment. Maybe someone else will have a suggestion.
If you have reason to believe that the installer forgot to remove one or more duplex receptacle fins, you have to get in the outlet boxes to fix the problem. Take off the covers to the outlets in the room. If you're lucky, there will be both red and black wires connected to the receptacle(s) with switched power. These are the receptacles where the fin should be removed.
If there is only black wires and no red wires, your next step is to find out how the installer connected the outlets to one another another. He could have used pigtails, using wire nuts to connect the "line" (wires coming into the outlet box) to the "load" (wires going to the next outlet). Or he could have daisy chained the outlets together, meaning both the the line and the load load is connected directly to the receptacle. If you find that the installer used pigtails, you can just look for the receptacles where both the top and bottom outlets are wired. This receptacle likely has your switched outlet. If they are daisy chained, you have your work cut out for you. I can't think of any other way than to start taking apart the outlets and testing the wires one by one.
If you find a receptacle that needs the fin removed, and there is a shared neutral, only take the hot fin off. If there is a neutral for both outlets, then take both fins off.
Safety note: Don't assume that all the wires in one box are of the same circuit. Test ALL the wires in the box before you go in there with your hands.
Best Answer
If I understand you right, you’re using a light and TV on a dimmer switch controlled Plug (outlet)
If so that is a extremely bad idea as this is totally unsuited for TVs, Stereos, Computers, and other equipment. Equipment of this type will likely kill the electronic switch (triac) inside the dimmer switch and will put the attached appliances into a condition where they are likely to be destroyed when the dimmer is set at anything less than 100% on.
Solutions:
You can if you want to keep the on off switch, remove the dimmer and put a regular light wall switch
The best would be to remove the Dimmer/light switch all together (After shutting power off from Breaker) and connect the switching wires together(NOT THE GROUNG)and cap, Then put a blank wall plate on.