An electronic device may be provided with a housing. The housing may have a periphery that is surrounded by peripheral conductive structures such as a segmented peripheral metal member. A segment of the peripheral metal member may be separated from a ground by a slot. An antenna feed may have a positive antenna terminal coupled to the peripheral metal member and a ground terminal coupled to the ground and may feed both an inverted-F antenna structure that is formed from the peripheral metal member and the ground and a slot antenna structure that is formed from the slot. Control circuitry may tune the antenna by controlling adjustable components that are coupled to the peripheral metal member. The adjustable components may include adjustable inductors and adjustable capacitors.