Consider a shortest closed geodesic $\gamma$ on the surface of length sys, and the normal exponential map of $\gamma$. Using the lower curvature bound, we obtain an upper bound on the total area as $\text{sys}\cdot \sinh(D)$ where $D$ is the diameter. This follows just by applying Rauch bounds on Jacobi fields (this is an ingredient in the proof of Toponogov). Therefore the systole is bounded below by $ \frac{\text{area}}{\sinh D}$ and the area is bounded below by Gauss-Bonnet. Furthermore the filling radius is bounded below by the 1/6 of the systole by Gromov's inequality. The least Gromov-Hausdorff distance to a graph is bounded below by the filling radius. We therefore get a quantitative lower bound and not merely nonexistence of Yamaguchi-type collapse. 

This proves that hyperbolic surfaces of curvature bounded below by $-1$ with diameter bounded above by $D$ cannot collapse so that a Gromov-Hausdorff limit is necessarily 2-dimensional.

For more details see https://arxiv.org/abs/1604.06782