We study estimation in parametric models. It is assumed that the variable of interest cannot be observed directly so a mixed case interval censoring model is used instead. The data consist of a sequence of times of inspection events and a mark variable. The mark variable indicates the endpoints of the inspection interval where the variable of interest is located. The main result is an extension of Wald's theorem for complete data to the censored case. The theorem provides sufficient conditions for consistency of the maximum likelihood estimates. The conditions obtained are much weaker compared to the standard Cramér-like regularity conditions.