  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 $\rho^W_p=p \proj{\psi^-}+(1-p){\one}/{4}$, we show
that there is a local model for projective measurements if and only if $p \le
1/K_G(3)$, 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 $p \lesssim 0.66$,
quite close to the current region of Bell violation, $p \sim 0.71$. We
generalize this result to arbitrary quantum states.
