A wireless electronic device may include antenna structures and antenna tuning circuitry. The device may include a display mounted within a housing. A peripheral conductive member may run around the edges of the display and housing. Dielectric-filled gaps may divide the peripheral conductive member into individual segments. A ground plane may be formed within the housing. The ground plane and the segments of the peripheral conductive member may form antennas in upper and lower portions of the housing. The antenna tuning circuitry may include switchable inductor circuits and variable capacitor circuits for the upper and lower antennas. The switchable inductor circuits associated with the upper antenna may be tuned to provide coverage in at least two high-band frequency ranges of interest, whereas the variable capacitor circuits associated with the upper antenna may be tuned to provide coverage in at least two low-band frequency ranges of interest.