The recent literature on "complex contagions" challenges Granovetter's classic hypothesis on the strength of weak ties and argues that, when the actors' choice requires reinforcement from several sources, it is the structure of strong ties that really matters to sustain rapid and wide diffusion. The paper contributes to this debate by reporting on a small-N study that relies on a unique combination of ethnographic data, social network analysis, and computational models. In particular, we investigate two rural populations of Indian and Kenyan potters who have to decide whether adopting new, objectively more efficient and economically more attractive, technical/stylistic options. Qualitative field data show that religious sub-communities within the Indian and Kenyan populations exhibit markedly different diffusion rates and speed over the last thirty years. To account for these differences, we first analyze empirically observed kinship networks and advice networks, and, then, we recreate the actual aggregate diffusion curves through a series of empirically-calibrated agent-based simulations. Combining the two methods, we show that, while single exposure through heterophilious weak ties were sufficient to initiate the diffusion process, large bridges made of strong ties can in fact lead to faster or slower diffusion depending on the type of signals circulating in the network. We conclude that, even in presence of "complex contagions", dense local ties cannot be regarded as a sufficient condition for faster diffusion.