For a broad class of high-T-c superconductor cuprates with nearly independent Cu-O planes, the values of the parameters defining the t-J model Hamiltonian are obtained from accurate ab initio electronic structure calculations on embedded cluster models. From the calculated t and J values a clear-cut correlation between T-c and these values emerges. Based on the ideas of resonating-valence-bond (RVB) theory, a possible interpretation of the origin of this correlation is discussed. The present work stresses the adequacy of the t-J model and strongly supports the idea of a pairing mechanism in cuprates mediated by magnetic interactions.