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Reliability is one of the crucial requirements of conductors used in resistive superconducting fault current limiters. Possible critical current degradation of a REBCO coated conductor during the limiter operation restrains broader application of devices based on these conductors. In this article the impact of an electrical stabilization layer on the conductor protection against overheating as well as its effect on electrical current evolution during the limitation period are investigated. For the experimental part of the study, a commercial copper stabilized REBCO coated conductor and a modified conductor without copper stabilization are compared. Extensive experimental work complemented by electro-thermal numerical modeling paved the way for studying the electrical and thermal effects separately. The numerical model assuming adiabatic conditions is sufficient to reproduce experimental results and predict the peak temperature for conductors with various stabilization layers in realistic conditions. Reduction of the conductor critical current caused by multiple current limitation pulses was studied using pulses with various durations. It was observed that the degradation due to temperature rise is gradual with the rate depending solely on the peak temperature. It is this quantity through which other parameters like the pulse duration, the thickness of stabilization and the peak current cause the tape damaging. [ABSTRACT FROM AUTHOR]