The current-carrying capacity (CCC), or ampacity, of highly-conductive, light, and strong carbon nanotube (CNT) fibers is characterized by measuring their failure current density (FCD) and continuous current rating (CCR) values. It is shown, both experimentally and theoretically, that the CCC of these fibers is determined by the balance between current-induced Joule heating and heat exchange with the surroundings. The measured FCD values of the fibers range from 107 to 109 A m-2 and are generally higher than the previously reported values for aligned buckypapers, carbon fibers, and CNT fibers. To the authors' knowledge, this is the first time the CCR for a CNT fiber has been reported. The specific CCC value (i.e., normalized by the linear mass density) of these CNT fibers are demonstrated to be higher than those of copper.