Biological invasions are a major threat to biodiversity, agriculture, and human health. Invasive populations can be the source of additional new introductions, leading to a self-accelerating process whereby invasion begets invasion. This phenomenon, coined bridgehead effect, has been proposed to stem from the evolution of higher invasiveness in a primary introduced population. There is, however, no conclusive evidence that the success of bridgehead populations stems from the evolution of increased invasiveness. Instead, we argue that a high frequency of secondary introductions can be explained by increased abundance in the bridgehead region or the topology of human transport networks. We outline the type of evidence and experiments that are needed to demonstrate adaptive evolution and higher invasion success of introduced populations.