We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating n noisy training data points in R^d by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound Ω(√(n/p)) of the interpolating neural network with p parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound Ω(n^1/d) for robust interpolation. Our results demonstrate a two-fold law of robustness: a) we show the potential benefit of overparametrization for smooth data interpolation when n=poly(d), and b) we disprove the potential existence of an O(1)-Lipschitz robust interpolating function when n=(ω(d)).