This article discusses narrative intelligence in the context of the evolution of primate (social) intelligence, and with respect to the particular cognitive limits that constrain the development of human social networks and societies. The Narrative Intelligence Hypothesis suggests that the evolutionary origin of communicating in a narrative format co-evolved with increasingly complex social dynamics among our human ancestors. This article gives examples of social interactions in non-human primates and how these interactions can be interpreted in terms of nonverbal narratives. The particular format of preverbal narrative that infants learn through transactions with others is important for the development of communication and social skills. A possible impairment of the construction of narrative formats in children with autism is discussed. Implications of the Narrative Intelligence Hypothesis for research into communication and social interactions in animals and robots are outlined. The article concludes by discussing implications for humane technology development.