Estimating the remaining useful life (RUL) of a bearing is required for maintenance scheduling. While the degradation behavior of a bearing changes during its lifetime, it is usually assumed to follow a single model. In this letter, bearing degradation is modeled by a monotonically increasing function that is globally non-linear and locally linearized. The model is generated using historical data that is smoothed with a local linear estimator. A neural network learns this model and then predicts future levels of vibration acceleration to estimate the RUL of a bearing. The proposed method yields reasonably accurate estimates of the RUL of a bearing at different points during its operational life.