The constant-sum property given in Oller et al. (2004) for censoring models justifies the use of a simplified likelihood to obtain the nonparametric maximum likelihood estimator of the lifetime distribution. In this paper we study the relevance of the constant-sum property in the identifiability of the lifetime distribution. We show that the lifetime distribution is not identifiable outside the class of constant-sum models. We also show that the lifetime probabilities assigned to the observable intervals are identifiable inside the class of constant-sum models. We illustrate all these notions with several examples.