A formal Starfleet proceeding to decide whether a commanding officer is still fit for duty, convened against a rapidly ageing Kirk in The Deadly Years.
A competency hearing is the procedure laid down in Starfleet regulations for removing a captain who can no longer do the job. It can be demanded when there is evidence of physical or mental unfitness, and the first officer must then convene a board and hear witnesses.
The procedure is shown in the Original Series episode The Deadly Years. Radiation at Gamma Hydra IV causes Kirk, Spock, McCoy and Scott to age decades in days. Kirk grows forgetful, repeating orders and signing the same report twice. Commodore Stocker, a passenger, insists that Spock hold a hearing, and Spock complies with great reluctance.
The crew's testimony leaves the board no choice, and Kirk is relieved. Stocker takes command and steers straight into the Romulan Neutral Zone, and the ship is saved only when a cured Kirk returns to the bridge.