In this paper register based family studies provide the motivation for studying a two-stage estimation procedure in copula models for multivariate failure time data. The asymptotic properties of the estimators in both parametric and semi-parametric models are derived, generalising the approach by Shih and Louis (Biometrics vol. 51, pp. 1384-1399, 1995b) and Glidden (Lifetime Data Analysis vol. 6, pp. 141-156, 2000). Because register based family studies often involve very large cohorts a method for analysing a sampled cohort is also derived together with the asymptotic properties of the estimators. The proposed methods are studied in simulations and the estimators are found to be highly efficient. Finally, the methods are applied to a study of mortality in twins.