The thermodynamic equilibrium state of crystalline materials is a single crystal; however, polycrystalline grain growth almost always stops before this state is reached. Although typically attributed to solute drag, grain-growth stagnation occurs, even in high-purity materials. Recent studies indicate that grain boundaries undergo thermal roughening associated with an abrupt mobility change, so that at typical annealing temperatures, polycrystals will contain both smooth (slow) and rough (fast) boundaries. Mesoscale grain-growth models, validated by large-scale polycrystalline molecular dynamics simulations, show that even small fractions of smooth, slow boundaries can stop grain growth. We conclude that grain-boundary roughening provides an alternate stagnation mechanism that applies even to high-purity materials.