This paper considers the competing risks model with two causes of death to analyze time-to-event data for a group of male mice exposed to three hundred roentgen radiation for 5–6 weeks. The analysis is based on the assumption that the parent distribution is the Xgamma distribution and the data are gathered using an adaptive Type-I progressively censored sample. Two estimation approaches are considered to complete the analysis: maximum likelihood and Bayesian methods. Besides acquiring the estimations of the model parameters, the estimations of the reliability and failure rate are also discussed. Both point and interval estimates using both estimation approaches are studied. In Bayesian estimations, the squared error loss function is used and the Markov Chain Monte Carlo technique is proposed to get samples from the joint posterior distribution. The various methods are compared using simulation studies to compare their performance. The mentioned radiation data set is investigated and the analysis showed the suitability of the competing risks model with Xgamma distribution to analyze the data and to estimate the reliability metrics.