Bearings are the most commonly used components in rotating machines and the ability to diagnose their faults and predict their remaining useful life (RUL) is critical for system maintenance. This paper proposes a smart system combined with a regression model to predict the RUL of bearings. The method converts the azimuth signal through low-pass filtering (LPF) and a chaotic mapping system, and uses Euclidean feature values (EFVs) to extract features in order to construct useful health indicators (HIs). In fault detection, the iterative cumulative moving average (ICMA) is used to smooth the HIs, and the Euclidean norm is used to find the time-to-start prediction (TSP). In terms of prediction, this paper uses a self-selective regression model to select the most suitable regression model to predict the RUL of the bearing. The dataset provided by the Center for Intelligent Maintenance Systems (IMS) is applied for performance evaluation; in comparison with previous research, better prediction results can be achieved by applying the proposed smart assessment system. The proposed system is also applied to the PRONOSTIA (also called FEMTO-ST) bearing dataset in this paper, demonstrating that acceptable prediction performance can be obtained.