In Proposition 7.4.12 of the book on metric spaces by Burago-Burago-Ivanov, it is mentioned that for compact metric spaces $X$ and $\{ X_n\}_{n=1}^{\infty}$, $X_n$ converges Gromov-Hausdorff to $X$ if and only if for every $\epsilon > 0$ there exists a finite $\epsilon$-net $S$ in $X$ and an $\epsilon$-net $S_n$ in each $X_n$ such that $S_n$ converges Gromov-Hausdorff to $S$.
Is it ever possible to show explicitly that such nets do not exist for some particular $X$ and $X_n$ and so that they do not converge Gromov-Hausdorff? What would be an example of where this can be done?