Wind turbines are required to meet grid interconnection codes for voltage ride-through, which mandate that they should not disconnect during a specified range of grid disturbances. One of the tasks of a turbine's design verification process is to ensure that it will be able to withstand any such event. This paper sets forth a powerful computational technique for design verification of Type-4 permanent-magnet synchronous generator-based turbines with fully-rated grid-side converters. The proposed method can provide assurance as to whether the currents and voltages will remain within acceptable bounds. To this end, reachability theory is employed to find sets that enclose all possible trajectories for the system states caused by a given set of unknown-but-bounded disturbances. The computation of the reachable sets is carried out by using on-the-fly linearization of the nonlinear system dynamics, zonotopes, and interval analysis.