We present a method for finding optimal partial solutions to overconstrained instances of the Disjunctive Temporal Problem (DTP). The solutions are optimal in that they satisfy a maximal number of constraints. While partial constraint satisfaction (PCS) has been commonly applied to finite-domain CSPs, its application in this setting is of particular interest, as temporal problems are traditionally solved using a meta-CSP approach, in which the constraints of the original problem become the variables of the meta-level problem. We show how to adopt nearly all previous pruning techniques in DTP solving for use with PCS, and provide results demonstrating their effectiveness. We also introduce an incremental technique that leads to substantially improved performance.