From what you describe its probably low voltage condition. The greater the load on a circuit the more the voltage drops. You need to use a multimeter to determine what the voltage is on that circuit when nothing is turned on. Connect the multimeter to a wall outlet on that circuit - not to the UPS outlets. It should be around 120 volts. Then measure the voltage again when the fault light is on. If the voltage is below 115 volts the fault light is most likely caused by the voltage drop.
Low voltage can be a real problem, however most consumer electronics will tolerate voltages as low as 110 volts. Your UPS will cut over to batteries if the voltage falls too much so you don't have to worry about anything plugged into the UPS. Fixing a low voltage condition can be complicated. It can be anything from a loose connection on the back of an outlet to a problem with the transformer at the street.