This contribution presents the results of numerical simulations in which a high-pier viaduct under lateral and vertical non-stationary spatially variable seismic ground motions is crossed by an articulated high-speed train at several speeds. The spectral representation method is employed to generate the required number of acceleration time-history series at each support location of the railway viaduct studied. Wave passage, local soil conditions and loss of coherency effects have been accounted for in the simulated ground-motion time histories, with several peak ground accelerations. Train and bridge responses have been obtained by means of a non-linear dynamic interaction multibody and finite element model. An analysis of train running safety indices has been carried out. The properties of earthquakes and train speeds which may cause problems to traffic safety are detailed, as is the train derailment probability. The numerical simulation outcomes show that it is not safe for the train to travel over the studied viaduct in earthquakes with peak ground acceleration equal to 0.03 g and for train speeds above 280 km/h.