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 ρ^W_p=p ψ^-+(1-p)/4, we show that there is a local model for projective measurements if and only if p < 1/K_G(3), where K_G(3) is Grothendieck's constant of order 3. Known bounds on K_G(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.