Truncated hierarchical B-splines have received significant attention in the isogeometric analysis community due to their compelling properties from a design perspective and because of their excellent numerical stability and potential for local refinement from an analysis perspective. Nonetheless, refinement techniques that can appropriately capture behavior for approximating solutions to some partial differential equations can lead to inaccurate solutions for other problems. Particularly, saddle point problems, including those of interest in fluid and electromagnetic analysis, must discretely adhere to the cohomological structure of the solution space for stability, and refinement methods that apply in other scenarios may not apply for these analyses. In this presentation, we give an overview of sufficient, locally-verifiable conditions that guarantee that the discrete de Rham sequence for (truncated) hierarchical B-splines is isomorphic to the infinite-dimensional de Rham sequence for a domain. We then present results in two and three dimensions numerically evaluating the stability properties of refinements that satisfy these conditions.