We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ρWp=p∣ψ−⟩⟨ψ−∣+(1−p)1/4, we show that there is a local model for projective measurements if and only if p⩽1/KG(3), where KG(3) is Grothendieck's constant of order 3. Known bounds on KG(3) prove the existence of this model at least for p≲0.66, quite close to the current region of Bell violation, p∼0.71. We generalize this result to arbitrary quantum states.