Pleiotropy has been suggested as a novel mechanism for stabilising cooperation in bacteria and other microbes. The hypothesis is that linking cooperation with a trait that provides a personal (private) benefit can outweigh the cost of cooperation in situations when cooperation would not be favoured by mechanisms such as kin selection. We analysed the theoretical plausibility of this hypothesis, with analytical models and individual-based simulations. We found that (1) pleiotropy does not stabilise cooperation, unless the cooperative and private traits are linked via a genetic architecture that cannot evolve (mutational constraint); (2) if the genetic architecture is constrained in this way, then pleiotropy favours any type of trait and not especially cooperation; (3) if the genetic architecture can evolve, then pleiotropy does not favour cooperation; and (4) there are several alternative explanations for why traits may be linked, and causality can even be predicted in the opposite direction, with cooperation favouring pleiotropy. Our results suggest that pleiotropy could only explain cooperation under restrictive conditions and instead show how social evolution can shape the genetic architecture.