Randomized smoothing, using just a simple isotropic Gaussian distribution, has been shown to produce good robustness guarantees against ℓ_2-norm bounded adversaries. In this work, we show that extending the smoothing technique to defend against other attack models can be challenging, especially in the high-dimensional regime. In particular, for a vast class of i.i.d. smoothing distributions, we prove that the largest ℓ_p-radius that can be certified decreases as O(1/d^1/2 - 1/p) with dimension d for p > 2. Notably, for p ≥ 2, this dependence on d is no better than that of the ℓ_p-radius that can be certified using isotropic Gaussian smoothing, essentially putting a matching lower bound on the robustness radius. When restricted to generalized Gaussian smoothing, these two bounds can be shown to be within a constant factor of each other in an asymptotic sense, establishing that Gaussian smoothing provides the best possible results, up to a constant factor, when p ≥ 2. We present experimental results on CIFAR to validate our theory. For other smoothing distributions, such as, a uniform distribution within an ℓ_1 or an ℓ_∞-norm ball, we show upper bounds of the form O(1 / d) and O(1 / d^1 - 1/p) respectively, which have an even worse dependence on d.