We show that the survival curve is identifiable in bivariate censored data problems under weaker independence assumptions than have commonly been made. The common assumption has been mutual independence of (T1,T2) and (Z1,Z2), where (T1,T2) is the true survival vector, (Z1,Z2) is a nuisance censoring vector, and bivariate right-censored data is observed. We show that the distribution of (T1,T2) is identifiable under weaker, conditional independence assumptions for distributions with full support. Bivariate survival analysis is a more powerful analysis tool than univariate analysis if multiple, possibly related, times are of interest. The mutual independence model has become popular as a nonparametric way of analyzing such data. Analysis of the bivariate problem and analogy with univariate models are used to show that the conditional independence model is more widely applicable as a general nonparametric model for bivariate survival data.