The authors prove quantitative versions of rigidity theorems for volume or diameter in the presence of a suitable lower bound on Ricci curvature. The model spaces here form a class of smooth manifolds with a warped product metric of a particular type. They show that if the volume or diameter are almost maximal, then the manifold is close to a warped product in the Gromov-Hausdorff sense. Among the applications are the splitting theorem for Gromov-Hausdorff limit spaces when the lower bound for Ricci curvature tends to 0, as well as Gromov’s conjecture that manifolds of almost positive Ricci curvature have almost nilpotent fundamental group.