Bogies are crucial for the safe operation of rail transit systems and usually work under uncertain and variable operating conditions. However, the diagnosis of bogie faults under variable conditions has barely been discussed until now. Thus, it is valuable to develop effective methods to deal with variable conditions. Besides, considering that the normal data for training are much more than the faulty data in practice, there is another problem in that only a small amount of data is available that includes faults. Concerning these issues, this paper proposes two new algorithms: (1) A novel feature parameter named spectral kurtosis entropy (SKE) is proposed based on the protrugram. The SKE not only avoids the manual post-processing of the protrugram but also has strong robustness to the operating conditions and parameter configurations, which have been validated by a simulation experiment in this paper. In this paper, the SKE, in conjunction with variational mode decomposition (VMD), is employed for feature extraction under variable conditions. (2) A new learning algorithm named weighted self-adaptive evolutionary extreme learning machine (WSaE-ELM) is proposed. WSaE-ELM gives each sample an extra sample weight to rebalance the training data and optimizes these weights along with the parameters of hidden neurons by means of the self-adaptive differential evolution algorithm. Finally, the hybrid method based on VMD, SKE, and WSaE-ELM is verified by using the vibration signals gathered from real bogies with speed variations. It is demonstrated that the proposed method of bogie fault diagnosis outperforms the conventional methods by up to 4.42% and 6.22%, respectively, in percentages of accuracy under variable conditions.