An electronic device may be coupled to an accessory such as a pair of earphones. The earphones may have noise cancellation features that may be implemented using noise cancellation circuitry in the earphones or in the electronic device. The earphones may have ear presence sensor structures that determine whether speakers in the earphones are present at the ears of a user. In one suitable embodiment, control circuitry in the earphones may be used to adjust noise cancellation circuitry in the earphones based on information from the ear presence sensor structures. For example, the control circuitry may deactivate noise cancellation circuitry in response to receiving information from the ear presence sensor structures indicating that the earphones have been removed from a user's ears. In another suitable embodiment, control circuitry in the electronic device may adjust noise cancellation circuitry in the electronic device based on information from the ear presence sensor structures.