Medicare's prospective payment system for long-term acute-care hospitals (LTCHs) provides modest reimbursements at the beginning of a patient's stay before jumping discontinuously to a large lump-sum payment after a prespecified number of days. We show that LTCHs respond to the financial incentives of this system by disproportionately discharging patients after they cross the large-payment threshold. We find this occurs more often at for-profit facilities, facilities acquired by leading LTCH chains, and facilities colocated with other hospitals. Using a dynamic structural model, we evaluate counterfactual payment policies that would provide substantial savings for Medicare.