The present paper introduces a Model Predictive Control (MPC) strategy for fully-constrained Cable-Driven Parallel Robots. The main advantage of the proposed scheme lies in its ability to handle cable tension limits explicitly. Indeed, the cable tension distribution is performed as an integral part of the main control architecture. This characteristic improves the safety of the system. Experimental results demonstrate this advantage addressing a typical pick-and-place task with two different scenarios: nominal cable tension limits and reduced maximum tension. Satisfactory tracking errors were obtained in the first scenario. In the second scenario, the desired trajectory escapes from the workspace defined by the new set of tension limits. The MPC is able to minimize the tracking error without violating the tension limits. Satisfying results were also obtained regarding robustness against uncertainties on the payload.