Thin films of a low molecular weight, nonvolatile liquid which are forced to spread on silicon wafers, rupture within minutes and dewet. Addition of long polystyrene chains (for which the liquid is a good solvent) up to a concentration of 10% does not change this behavior qualitatively. In contrast we find that the unbroken uniformity of the films may be preserved for many months or longer by a polystyrene brush attached to the silicon, together with some unattached polystyrene in the liquid. By varying the molecular weights and concentrations of the unattached chains within the film, we were able to establish a stability diagram in this system which shows that suppression of rupture is only observed at free-polymer concentrations which exceed the overlap concentration. This suggests that the effect may arise from the formation of an entanglement network between the free chains (within the liquid film) and the surface-tethered brush molecules.