Hello Patrick,
If the height map has high frequency detail, it is likely that the mesh will not have a dense enough geometry to properly model the finer detail in the texture.
To mitigate this issue, you may use a higher Tessellation factor: this will subdivide the base mesh by a greater amount, resulting in finer geometry at the cost of some performance.
In Designer, you can find this setting in the 3D View panel: open the Material menu, then select the Default > Edit to display the shader properties in the Properties panel. The Tessellation factor property should be in the Height section. In Painter, this is managed in the Shader settings (see below).
That being said, my recommendation is using the Height output for the main features of the shape/silhouette (i.e. low to mid frequency detail), and add surface micro details (i.e. high frequency detail) in the Normal output only
Keep in mind some texture stretching is to be expected where Height changes abruptly and the displaced geometry is thus stretched. Therefore, keep the transitions in the height map smooth/beveled enough to keep a clean result when using displacement.
To learn more about managing this technique in Painter, take a look at the Displacement and tessellation section of the Shader settings page in Painter's documentation. The page also includes a mention of the Height to normal filter which may prove useful.
This short tutorial from Xolotl Studios (shout out for the great content!) is also relevant.
Best regards.