The thermal stability of superconducting YBCO-coated conductors subject to over-currents are analysed. We have studied the effect of DC and AC over-current pulses in Cu-stabilized and non-stabilized coated conductors by measuring the electric field and temperature profiles of these conductors immersed in liquid nitrogen. Current pulses of short duration of about 90 ms and long duration of a few seconds were applied to the samples. Three different cooling regimes of liquid nitrogen-convection, nucleate boiling and film boiling-were observed and their influence on the recovery time of superconductivity in the coated conductors after the over-current pulses has been analysed. We have studied the recovery behaviour under two different conditions, in which the current was set to zero and to the operating current after the current pulses. These experiments simulated the conditions during an over-current situation in different electric power applications with special attention given to the behaviour of these coated conductors acting as in-fault current limiters.