The failure of individual multiwall carbon nanotubes during field emission and two-probe characterization was studied in a scanning electron microscope on emitters grown by chemical vapor deposition. The failures induced by high currents during two-probe measurements led to a sundering of the tube in or near its middle, which is consistent with a local evaporation due to resistive heating. Conversely, failures during field emission occurred at or near the substrate-emitter contact. We show that the degradation is due to mechanical failure of the contact at low applied fields, and to resistive heating (probably enhanced by the mechanical stress) at high emitted currents.