Summary: We analyse the identifiability of the number of components in \(k\)-variate, \(M\)-component finite mixture models in which each component distribution has independent marginals, including models in latent class analysis. Without making parametric assumptions on the component distributions, we investigate how one can identify the number of components from the distribution function of the observed data. When \(k\geq 2\), a lower bound on the number of components (\(M\)) is non-parametrically identifiable from the rank of a matrix constructed from the distribution function of the observed variables. Building on this identification condition, we develop a procedure to estimate a lower bound on the number of components consistently.