In many matching markets bargaining determines who matches with whom and on what terms. We experimentally investigate allocative efficiency and how subjects' payoffs depend on their matching opportunities in such markets. We consider three simple markets. There are no information asymmetries, subjects are patient and a perfectly equitable outcome is both feasible and efficient. Efficient perfect equilibria of the corresponding bargaining game exist, but are increasingly complicated to sustain across the three markets. Consistent with the predictions of simple (Markov perfect) equilibria, we find considerable mismatch in two of the markets. Mismatch is reduced but remains substantial when we change the nature of bargaining by moving from a structured experimental protocol to permitting free-form negotiations, and when we allow players to renege on their agreements. Our results suggest mismatch is driven by players correctly anticipating that they might lose their strong bargaining positions, and by players in weak bargaining positions demanding equitable payoffs.