Border carbon adjustments (BCAs) have been suggested as a measure to reduce carbon leakage in the presence of unilateral climate policies and/or to enforce cooperative climate agreements. In an intra-industry trade model, this paper studies whether and under which conditions a sequence of escalating threats of implementing BCA-measures could be successful in enforcing a fully cooperative agreement. We start from a situation where moving from non-cooperative production-based carbon taxes to a socially optimal tax is not attractive to the environmentally less concerned country. We then test whether the threat of imposing BCA-measures, in the form of import tariffs or, additionally, supplemented by export rebates, will enforce cooperation. We show that import tariffs are the least distortionary policy instrument but the weakest threat, and import tariffs with a full export rebate is the most distortionary instrument if implemented but the most effective threat to enforce cooperation. In an escalating penalty game, we determine the subgame-perfect equilibrium path along which threats are deterrent and credible. We show that BCA-measures help to enforce cooperation, reduce global emissions and are welfare improving if they need to be implemented. However, whenever full cooperation would generate the highest global welfare gains, BCAs fail to establish cooperation, a version of the paradox of cooperation, as proposed by Barrett (1994).