The hypobaric decompression sickness data study was conducted by the National Aeronautics and Space Administration to investigate the risk of decompression sickness in hypobaric environments. The quantity of interest is the time to onset of grade IV venous gas emboli, which was mixed case interval censored because of measurement limitations. In the study, some subjects participated in multiple experiments, leading to repeated and correlated measurements on those subjects. In addition, it has been suggested that some subjects had a much lower risk of developing grade IV venous gas emboli than others, i.e. those subjects were "immune" from the event of interest (or 'cured'). We propose to use two-part models, where the first part describes the probability of cure and the second part describes the survival for susceptible subjects. We use two random effects to account for the correlated nature of measurements. A leverage bootstrap approach is proposed for model diagnosis. A simulation study shows satisfactory performance of the estimation and diagnosis approaches proposed. Model estimation and evaluation of the hypobaric decompression sickness data are carefully investigated. Copyright (c) 2010 Royal Statistical Society.