Considering the competitive failure that exists in the operation of industrial systems, including degradation failure and sudden failure, this paper presents a reliability assessment method based on the three-parameter Weibull distribution and the Wiener process. Nonlinear functions are proposed to establish the relationship between the different failure processes, and the reliability model is derived. The Metropolis-Hastings (MH) sampling algorithm of the Monte Carlo Markov Chain (MCMC) method is employed to estimate the parameters in this study. The reliability assessment results are obtained by the real degradation samples. The results show that the reliability model incorporating the three-parameter Weibull distribution produces more comprehensive and dependable results. Furthermore, MH sampling can solve the issues of complex likelihood functions that cannot directly obtain parameter evaluation results. Additionally, this paper analyzes the sensitivity of the model parameters, thereby offering theoretical support for enhancing the safe operation of the system.