Polymers grafted or otherwise immobilized at surfaces are often used to stabilize particle suspensions in situations where nonspecific colloidal forces or specific chemical bonds act to coagulate the particles. Although the motivation is clear, criteria for steric stabilization are not well established for environments that contain large amounts of free polymer which can induce strong attraction through depletion forces. To examine these criteria, we have performed micromechanical tests of adhesion between pairs of large vesicles with membranes made from mixtures of phospholipids and phospholipids grafted with poly-(ethylene oxide) PEO chains in aqueous solutions of free (nonadsorbing) PEO chains. Solutions were made with two free polymer fractions (N-p approximate to 114 and 444 monomers in average length) over a large range of volume fractions between 0.01 and 0.2. Three graft chain lengths (N-g approximate to 45, 114, and 273 monomers) and several surface densities were used to extend the scale of surface roughness from similar to 0.5 nm for smooth bilayers to nearly 16 nm for the most prominent graft. On the basis of established theory, the grafted polymers were expected to form structures that varied from dilute noninteracting chains to semidilute marginal brushes with mean volume fractions of up to 0.06. Surprisingly, we found a consistent pattern of interaction for all graft lengths and surface densities; i.e., there was no adhesion below a threshold level of free polymer concentration and adhesion energy increased strongly with free polymer concentration above the threshold. Moreover, there was no tendency toward stabilization even at the highest volume fractions of 0.15-0.2. When both surfaces carried grafts (symmetric interactions), the stabilization threshold was proportionally higher than when one of the two surfaces was bare (asymmetric interactions). Significantly, these features were independent of free and graft polymer molecular weights. Correlation of the energy scale for free polymer depletion at the threshold to the graft surface density and symmetry of the steric interaction implies a simple criterion for stabilization of surfaces grafted with homopolymers in semidilute solutions of nonadsorbing free polymer and good solvent.