We discuss a new class of ignorable non-monotone missing data models-the randomized monotone missingness (RMM) models. We argue that the RMM models represent the most general plausible physical mechanism for generating non-monotone ignorable data. We show that there exists ignorable missing data processes that are not RMM. We argue that it may therefore be inappropriate to analyse non-monotone missing data under the assumption that the missingness mechanism is ignorable, if a statistical test has rejected the hypothesis that the missing data process is RMM representable. We use RMM models to analyse data from a case-control study of the effects of radiation on breast cancer.