This paper studies the robustness properties of a class of saddle-point dynamics for linear programming. This dynamics is distributed over a network in which every node controls one component of the optimization variable. In this multi-agent setting, communication noise, computation errors, and mismatches in the agents' knowledge about the problem data all enter into the dynamics as unmodeled disturbances. We show that the saddle-point dynamics is integral input-to-state stable and hence robust to disturbances of finite energy. This result also allows us to establish the robustness of the dynamics when the communication graph is recurrently connected because of link failures. Several simulations illustrate our results.