An electronic device such as a portable electronic device may have an antenna and associated wireless communications circuitry. A sensor such as a proximity sensor may be used to detect when the electronic device is in close proximity to a user's head. Control circuitry within the electronic device may be used to adjust radio-frequency signal transmit power levels. When it is determined that the electronic device is within a given distance from the user's head, the radio-frequency signal transmit power level may be reduced. When it is determined that the electronic device is not within the given distance from the user's head, proximity-based limits on the radio-frequency signal transmit power level may be removed. Data may be gathered from a touch sensor, accelerometer, ambient light sensor and other sources for use in determining how to adjust the transmit power level.