An electronic device may be provided with antenna structures. Proximity sensors and other sensors may be used in determining how the electronic device is being operated. Wireless circuitry such as a radio-frequency transmitter associated with a cellular telephone communications band, a wireless local area network band, or other communications band may be used in transmitting radio-frequency signals through the antenna structures at a transmit power. Control circuitry may adjust the wireless circuitry to ensure that the transmit power is capped at a maximum transmit power. The maximum transmit power may be adjusted dynamically by the control circuitry based on data from the proximity sensors, data from a magnetic sensor that detects whether a cover is present on the device, a connector sensor that detects whether the device is coupled to a dock or other accessory, and other sensors.