Damage identification and localization in railway bridges is a widely studied topic. Strain, displacement, or acceleration sensors installed on the bridge structure are normally used to detect changes in the global behavior of the structure, whereas approaches like ultra-sonic testing, acoustic emission, and magnetic inspection are used to check a small portion of structure near localized damage. The aim of this paper is to explore another perspective for monitoring the structural status of railway bridges, i.e., to detect structural damage from the dynamic response of the train transiting the bridge. This approach can successfully be implemented in the case of resonant bridges, thanks to the high level of acceleration generated, but its application becomes more challenging when the excitation frequencies due to train passage do not excite the first mode of vibration of the bridge. The paper investigates the feasibility of the method in the latter case, through numerical simulations of the complete train-track-bridge system. Accelerations on axleboxes and bogies are processed through suitable algorithms to detect differences arising when the train crosses a defective bridge or a healthy one. The results outline the main operational parameters affecting the method, the best placement for sensors, and the best frequency range to be considered in the signal processing, also addressing the issues that are related to track irregularity. Good performance can be achieved in the case of short bridges, but a few practical issues must be tackled before the method could be tested in practice.