Norway and ship rats (Rattus norvegicus and Rattus rattus) have invaded many habitats of conservation value worldwide, but Norway rats are widely assumed to be less of a threat to tree-nesting biota than are ship rats because they are less adept at climbing. We tested this assumption by measuring the capabilities of wild-caught captive rats of both species in reaching food rewards above ground, placed at fixed sites of increasingly difficult access. We confirmed that Norway rats were much slower and less agile than ship rats, but could in fact, given enough time, reach the same height above ground, run across the same thin ropes fixed at both ends and climb a real tree. However, they were more easily defeated by obstacles, more dependent on the availability of footholds, more vulnerable to falls, and those of >200 g body weight were significantly less likely to reach food rewards at the unsupported ends of small branches. We conclude that Norway rats seldom forage above ground, not because they cannot climb but because arboreal foraging is more risky and less likely to be rewarding for them.