An electronic device may have electronic device housing structures in which electronic components such as integrated circuits and connectors may be mounted. The electronic device housing structures may include an upper housing having a planar upper surface member and four perpendicular housing sidewall structures. The housing sidewall structures of the upper housing may have edges that form a rectangular opening with curved corners. A lower housing may have structures forming a rectangular lip that is configured to be received within the rectangular opening in the upper housing. Engagement structures such as inwardly protruding hook structures on the upper housing and snap structures on the rectangular lip may be used in attaching the upper and lower housings. The snap structures may each have a rectangular main opening and lateral extension portions that extend the width of the main opening along the edge of the lip.