Decades of research have led to numerous insights in modeling the impact of stresses and rock properties on hydraulic fracture height growth. However, the conditions under which weak horizontal interfaces are expected to impede height growth remain for the most part unknown. We have developed an experimental study of the impact of weak horizontal discontinuities on hydraulic fracture height growth, including the influences of (1) abrupt stress contrasts between layers, (2) material fracture toughness, and (3) contrasts of stiffness between the reservoir and bounding layers. The experiments are carried out with an analog three-layered medium constructed from transparent polyurethane, considering toughnesses resisting vertical fracture growth. There are four observed geometries: containment, height growth, T-shape growth, and the combination of height growth and T-shape. Results are developed in a parametric space embodying the influence of the horizontal stress contrast, vertical stress, and horizontal barrier stress contrast, as well as the fluid pressure. The results indicate that these cases fall within distinct regions when plotted in the parametric space. The locations in the parametric space of these regions are strongly impacted by the vertical fracture toughness: Increasing the value of the vertical interface fracture toughness leads to a suppression of height growth in favor of containment and T-shaped growth. Besides providing detailed experimental data for benchmarking 3D hydraulic fracture simulators, these experiments show that the fracture height is substantially less than would be predicted in the absence of the weak horizontal discontinuities.