Degradation signals are sensor-based signals that are correlated with degradation processes of engineering components. In this paper, we present a flexible modeling framework for characterizing degradation signals that can only be observed up to a prespecified failure threshold. The underlying assumption of this framework is that the engineering components degrade according to a similar trend, referred to as the common shape function, but at different degradation rates. Under this assumption, the degradation signals of different components are synchronized using a random time warping process, which transforms the common trend function into degradation processes that progress at varying rates. Our primary objective is to obtain real-time predictions for the residual lifetime of components deployed in the field. In the initial step, the historical degradation signals are used to recover the distribution of the degradation processes under the assumptions of the proposed time warping model. Next, the distribution of the degradation process is updated using the signal(s) of partially degraded component(s). The updated model is then utilized to predict the residual lifetime distributions of these components. We test the performance of our methodology using vibration-based degradation signals from a rotating machinery experiment and simulated degradation signals. Additional information and codes are available as supplementary material online.