Summary: In many medical studies, individuals are seen periodically, at a set of pre-scheduled clinical visits. In such cases, when the outcome of interest is the occurrence of an event, the corresponding times are only known to fall within an interval, formed by the times of two consecutive visits. Such data are called interval censored. Most methods for the analysis of interval-censored event times are based on a simplified likelihood function which relies on the assumption that the only information provided by the censoring intervals is that they contain the actual event time (i.e. non-informative censoring). In this simulation study, the performance of parametric models for interval-censored data when individuals miss some of the pre-scheduled visits completely at random (MCAR), at random (MAR) or not at random (MNAR) was assessed comparing also with a simpler approach that is often used in practice. A sample of HIV-RNA measurements and baseline covariates of HIV-1 infected individuals from the CASCADE study is used for illustration in an analysis of the time between the initiation of antiretroviral treatment and viral load suppression to undetectable levels. Results suggest that parametric models based on flexible distributions (e.g. generalised Gamma) can fit such data reasonably well and are robust to irregular visit times caused by an MCAR or MAR mechanism. Violating the non-informative censoring assumption though, leads to biased estimators with the direction and the magnitude of the bias depending on the direction and the strength of the association between the probability of missing visits and the actual time-to-event. Finally, simplifying the data in order to use standard survival analysis techniques, can yield misleading results even when the censoring intervals depend only on a baseline covariate.