We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin–Veretennikov–Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift u and show that for any $\alpha <0$, there exists a velocity field $u \in L^\infty_t C^\alpha_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $\alpha \geq 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry–Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain Hölder regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions.