Silver nanowire transparent electrodes have shown considerable potential to replace conventional transparent conductive materials. However, in this report we show that Joule heating is a unique and serious problem with these electrodes. When conducting current densities encountered in organic solar cells, the average surface temperature of indium tin oxide (ITO) and silver nanowire electrodes, both with sheet resistances of 60 ohms/square, remains below 35 °C. However, in contrast to ITO, the temperature in the nanowire electrode is very non-uniform, with some localized points reaching temperatures above 250 °C. These hotspots accelerate nanowire degradation, leading to electrode failure after 5 days of continuous current flow. We show that graphene, a commonly used passivation layer for these electrodes, slows nanowire degradation and creates a more uniform surface temperature under current flow. However, the graphene does not prevent Joule heating in the nanowires and local points of high temperature ultimately shift the failure mechanism from nanowire degradation to melting of the underlying plastic substrate. In this paper, surface temperature mapping, lifetime testing under current flow, post-mortem analysis, and modelling illuminate the behaviour and failure mechanisms of nanowires under extended current flow and provide guidelines for managing Joule heating.