We estimate the depairing current of superconducting nanowire single-photon detectors (SNSPDs) by studying the dependence of the nanowires' kinetic inductance on their bias current. The kinetic inductance is determined by measuring the resonance frequency of resonator-style nanowire coplanar waveguides both in transmission and reflection configurations. Bias current dependent shifts in the measured resonant frequency correspond to the change in the kinetic inductance, which can be compared with theoretical predictions. We demonstrate that the fast relaxation model described in the literature accurately matches our experimental data and provides a valuable tool for determination of the depairing current. Accurate measurement of the depairing current is critical for nanowire quality analysis, as well as modeling efforts aimed at understanding the detection mechanism in SNSPDs.