Vesicles prepared in water from a series of diblock copolymers-"polymersomes"-are physically characterized. With increasing molecular weight (M) over bar (n), the hydrophobic core thickness for self-assembled bilayers of poly(ethylene oxide) -polybutadiene increases up to similar to20 nm, which is considerably greater than any previously studied lipid or polymersome system. Micromanipulation of vesicles demonstrates an interface-dominated elasticity that is independent of (M) over bar (n). Furthermore, membrane stability as defined by the maximal areal strain increases with (M) over bar (n), approaching a universal limit predicted by mean-field ideas and set by the interfacial tension. Nonlinear responses and memory effects also emerge with increasing (M) over bar (n), indicating the onset of chain entanglements at higher (M) over bar (n). The results highlight the interfacial limits and transitions to bulk responses of self-assemblies.