In order to understand the ecological effects of climate change it is essential to forecast suitable areas for species in the future. However, species’ ability to reach potentially suitable areas is also critical for species survival. These ‘range-shift’ abilities can be studied using life-history traits related to four range-shift stages: emigration, movement, establishment, and proliferation. Here, we use the extent to which species’ ranges fill the climatically suitable area available (‘range filling’) as a proxy for the ability of European mammals and birds to shift their ranges under climate change. We detect which traits associate most closely with range filling. Drawing comparisons with a recent analysis for plants, we ask whether the latitudinal position of species’ ranges supports the assertion that post-glacial range-shift limitations cause disequilibrium between ranges and climate. We also disentangle which of the three taxonomic groups has greatest range filling. For mammals, generalists and early-reproducing species have the greatest range filling. For birds, generalist species with high annual fecundity, which live longer than expected based on body size, have the greatest range filling. Although we consider traits related to the four range-shift stages, only traits related to establishment and proliferation ability significantly influence range filling of mammals and birds. Species with the greatest range filling are those whose range centroid falls in the latitudinal centre of Europe, suggesting that post-glacial range expansion is a leading cause of disequilibrium with climate, although other explanations are also possible. Range filling of plants is lower than that of mammals or birds, suggesting that plants are more range-limited by non-climatic factors. Therefore, plants might be face greater non-climatic restraints on range shifts than mammals or birds.