The position and velocity information of highspeed trains (HSTs) are essential to passenger safety, operational efficiency, and maintenance, for which an accurate navigation system is required. In this paper, we propose a two-stage federated Kalman filter (TS-FKF) for an HST navigation system that uses multi-sensors, such as tachometer, inertial navigation system, differential GPS, and RFID, with a feedback scheme. However, the FKF with a feedback scheme often shows severe performance degradation in the presence of undetected large sensor errors. Tachometers often have large slip or slide errors during the train's acceleration, deceleration, and moving along a curved railway, and there are significant performance differences between different sensors. To make the proposed system robust to these errors, we propose a slip and slide detection algorithm for the tachometer and an adaptive information-sharing algorithm to deal with a large tachometer error and performance difference between sensors. We provide theoretical analysis and simulation results to demonstrate the performance of the proposed navigation system with the proposed algorithms.