Overcurrents in the power grid can have a magnitude of up to 20 times or higher than the rated current. This may cause problems and permanent damage to electrical equipment in the grid. High temperature superconducting (HTS) tapes are known to be sensitive to currents much larger than their critical current. In this light, it is important to investigate the response of HTS tapes and cable conductors to overcurrents several times the critical current. A number of experiments have been performed on HTS tapes and cable conductors, with currents up to 20 times the critical current. During overcurrent experiments, the voltage, and the temperature were measured as functions of time in order to investigate the dynamic behavior of the HTS tape and cable conductor. After each experiment, damage to the superconductors was assessed by measuring the critical current. Preliminary results show that within seconds an HTS tape (critical current=17 A) heats above room temperature with an overcurrent larger than 140 A. Similar overcurrent experiments showed that a HTS cable conductor could sustain damage with overcurrents exceeding 10 times the critical current of the cable conductor.