Handheld electronic devices are provided that contain wireless communications circuitry. The wireless communications circuitry may include antenna structures. To accommodate manufacturing variations, the antenna structures and handheld electronic devices may be characterized by performing measurements such as antenna performance measurements. Appropriate antenna adjustments may be made during manufacturing of a handheld electronic device based on the characterizing measurements. An antenna may be formed using an inverted-F design in which an antenna flex circuit is mounted to a dielectric antenna support structure. Cavities in the support may be selectively filled with dielectric material and dielectric patches may be added to the antenna flex circuit to adjust the dielectric loading of the antenna. The length of a ground return path in the antenna may be adjusted by appropriate positioning of an electrical connector within the ground return path.