We discuss some aspects of swampland constraints — especially the swamp-land distance conjecture — in a large number of space-time dimensions D. We analyze Kaluza-Klein (KK) states at large D and find that some KK spectra possess an interesting dependence on D. On the basis of these observations we propose a new large dimension conjecture. We apply it to KK states of compactifications to anti-de Sitter backgrounds where it predicts an upper bound on the dimension of space-time as a function of its characteristic radius. We also apply our conjecture to black hole spacetimes, whose entropies have a D-dependence very similar to that of the KK spectrum.