Scholars in the field of supply chain management have started to embrace the idea of simultaneous cooperation and competition (coopetition) in supplier networks but have mainly looked at coopetition from a structural perspective. In this article, we complement the structural view with a paradox perspective to investigate the competitive tensions that evolve when buying firms are trying to engage both forces simultaneously in their relations with core suppliers. Our comparative case study of four major carmakers reveals different strategies buying firms use to manage coopetition in their supplier networks, the responses they trigger from their suppliers, and the resulting paradoxical tensions. Our inductive analysis reveals that irrespective of the managing approach chosen, the coopetition capabilities of the buying firm determines whether negative tension dynamics can be avoided. At the core of such coopetition capabilities are evaluative capabilities allowing the buyer to provide cost improvement suggestions to suppliers for the sake of joint value creation and to control the division of value appropriation through a deeper understanding of the supplier's cost structures. By highlighting the nature of coopetition capabilities as organizational capabilities, we also contribute to paradox research, going beyond its current focus on the individual cognitions of managers.