We consider a two‐stage model of R&D/Cournot competition with isoelastic demand satisfying the cost paradox (i.e., that equilibrium profits increase with unit cost). The R&D process has a binary structure, with spillover effects. We provide a negative answer to the question in the title: Under noncooperative R&D, firms will conduct R&D for a broad parameter range, despite the presence of the cost paradox, as a result of being caught in a prisoner's dilemma. A second‐best social planner is shown to have a higher propensity for R&D than the noncooperative scenario. However, if firms engaged in any of the known R&D cooperation scenarios, the answer to the question in the title would become affirmative. It follows that R&D cooperation leads to lower producer and consumer surpluses. This constitutes a major departure from the conclusions of the standard R&D model. Therefore, R&D cooperation in such environments should not receive favorable antitrust treatment.