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The iterated Johnson bound is the best known upper bound on the size of an error-correcting code in the Grassmannian Gq(n,k). The iterated Schönheim bound is the best known lower bound on the size of a covering code in Gq(n,k). We prove that both bounds are asymptotically attained for fixed k and fixed radius, as n approaches infinity. Our methods rely on results from the theory of quasi-random hypergraphs which are proved using probabilistic techniques. We also determine the asymptotics of the size of the best Grassmannian codes and covering codes when n-k and the radius are fixed, as n approaches infinity.