It can. Saturated soil may slow dispersal, stormwater may temporarily raise a pump chamber, or an outage may leave stored wastewater waiting when power returns. Keep water use low and watch the labeled light. If the level falls and the light clears, record the time, rain, and household use instead of treating the event as meaningless.
Call within a day or two if the alarm stays on, returns with ordinary water use, or follows most storms. The provider should check how water is getting in and whether the pump and floats work, then look at the disposal area. In Maury's karst, repeated wet-weather alarms often trace back to saturated shallow soil that cannot disperse fast enough.
Do not pump a tank merely because the ground is saturated. Federal flood guidance warns that an empty or heavily pumped tank can move or float in saturated soil. A provider should decide whether controlled pumping is safe and useful for the exact tank, water level, and failure mode.