The contributions to the spin relaxation in copper (Cu) nanowires are quantified by carefully analyzing measurements of both charge and spin transport in lateral spin valves as a function of temperature and thickness. The temperature dependence of the spin-flip scattering solely arises from the scattering with phonons, as in bulk Cu, whereas we identify grain boundaries as the main temperature-independent contribution of the defects in the nanowires. A puzzling maximum in the spin diffusion length of Cu at low temperatures is found, which can be explained by the presence of magnetic impurities. The results presented here suggest routes for improving spin transport in metallic nanostructures, otherwise limited by confinement effects.