Longstanding mechanistic questions about the role of protecting osmolyte trimethylamine N-oxide (TMAO) that favors protein folding and the denaturing osmolyte urea are addressed by studying their effects on the folding of uncharged polymer chains. Using atomistic molecular dynamics simulations, we show that 1 M TMAO and 7 M urea solutions act dramatically differently on these model polymer chains. Their behaviors are sensitive to the strength of the attractive dispersion interactions of the chain with its environment: when these dispersion interactions are sufficiently strong, TMAO suppresses the formation of extended conformations of the hydrophobic polymer as compared to water while urea promotes the formation of extended conformations. Similar trends are observed experimentally for real protein systems. Quite surprisingly, we find that both protecting and denaturing osmolytes strongly interact with the polymer, seemingly in contrast with existing explanations of the osmolyte effect on proteins. We show that what really matters for a protective osmolyte is its effective depletion as the polymer conformation changes, which leads to a negative change in the preferential binding coefficient. For TMAO, there is a much more favorable free energy of insertion of a single osmolyte near collapsed conformations of the polymer than near extended conformations. By contrast, urea is preferentially stabilized next to the extended conformation and thus has a denaturing effect.