The Navascués–Pironio–Acín (NPA) hierarchy provides increasingly tighter semidefinite outer approximations to the set of quantum correlations, while its dual programs give sum-of-squares certificates that can establish exact quantum bounds for Bell inequalities and yield self-testing relations. We show that no finite level of the NPA hierarchy can exactly characterize the set of quantum correlations, even in the simplest Bell scenario. For the symmetric doubly tilted CHSH inequality, the maximal quantum violation vanishes cubically as the tilting parameter approaches its critical value, while the error of every fixed NPA level relative to this violation diverges. We construct a family of two-qubit strategies for which the gap between the exact quantum bound and the attained Bell value, divided by the quantum violation, tends to the Motzkin polynomial in rescaled measurement angles. If the relative error remained bounded, the fixed-level NPA certificates would yield a sum-of-squares representation of this limiting polynomial plus a nonnegative constant, which is impossible.