In this work we study the influence of the thermal contact resistance on the temperature of a typical nanostripe used in current induced magnetic domain wall movement or depinning. The thermal contact resistance arises from an imperfect heat transport across the interface between the metallic ferromagnetic nanostripe and the substrate. We show that this parameter, which is likely non-zero in any experimental device, increases the temperature in the nanostripe considerably. When the current is injected in the nanostripe in nanosecond long pulses, the larger temperature also implies a reduction of the effective current density delivered by the pulse generator. Both, the thermal contact resistance and the dynamic response of the pulse generator, are usually neglected in theoretical estimations of the influence of spin transfer torque on domain wall displacement and depinning. Here we show that only if the thermal contact resistance and the electric resistivity of the ferromagnetic nanostripe are optimized to the best values reported in the bibliography, the Joule heating may not be so crucial for current densities of the order of 108 A/cm2. Also, the use of physical constrictions (notch) to pin the magnetic domain wall may complicate the interpretation of the results as they always come together with relevant thermal gradients.