An electronic device may have a display with a backlight. The backlight provides backlight illumination for an array of pixels that is displaying images. The backlight may include an array of cells. Each cell may contain a light source with one or more light-emitting diodes and a cavity reflector that reflects light from the light source outwardly through a diffuser for use in forming the backlight illumination. The light sources may be mounted to a printed circuit. Support posts on the printed circuit may be used to maintain the diffuser at a fixed distance from the printed circuit. Sensors on the printed circuit may be used to measure separation distances between the printed circuit and diffuser. Adjustments to pixel gain profiles can be made based on the measured separation distances.