Their can be alot of factors causing this.
1. HOW many rooms are connected to this 15 amp load. I have seen houses that run 4 bedrooms (wall outlets and lights) on one circuit.
2. How far is the run from the panel. There is a thing called voltage drop that can adversely affect amperage draws. I saw this first hand last week. We had a commercial building with a 300 foot run of #12 wire placed on a 20 amp breaker. When they tried to run a 1-1/2 HP motor, it tripped the circuit. Nothing else was run on the wire. I placed an amp meter on the load and found the motor trying to pull 48 amps out of the #12 wire to start the motor. My electronic voltage drop test meter showed me this circuit was having an 18% voltage drop over this distance. On a 120 volt circuit that equated to a 21.6 volt drop, therefore this so called 120 volt circuit was actually only supplying 98.4 volts under load. With that low of voltage, the amperage draw went thru the roof while this motor tried to start and operate. I replaced the wire with the right size for a 5% voltage drop, and the motor fired up and run without any more issues.
3. In new construction, a window A/C units REQUIRES a dedicated, separate 20 amp home run. It cant be used to operate any other device.
4. It can be normal for the lights to dim MOMENTARILY when a large load is started, but they should return to full brightness in a second or two. If they don't return to full brightness, you have a problem that NEEDS to be looked at. Just because your breaker doesn't trip doesn't mean the circuit isn't overloaded. I have physically found wires that are HOT to the touch even though the breaker hasn't tripped. These circuits are overloaded and can cause a fire.
if in doubt.... call an electrical contractor. For a few hundred $$$ you may just save your house.