  We prove via explicitly constructed initial data that solutions to the
gravity-capillary wave system in $\mathbb{R}^3$ representing a 2d air-water
interface immediately fails to be $C^3$ with respect to the initial data if the
initial data $(h_0, \psi_0) \in H^{s+\frac12} \otimes H^{s}$ for $s<3$. Similar
results hold in $\mathbb{R}^2$ domains with a 1d interface. Furthermore, we
discuss the illposedness threshold for the pure gravity water wave system.
