If a pure simplicial complex is partitionable, then its h-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex increment , we construct a complex Gamma superset of increment of the same dimension such that both Gamma and the relative complex (Gamma , increment ) are partitionable. This allows us to rewrite the h-vector of any pure simplicial complex as the difference of two h-vectors of partitionable complexes, giving an analogous interpretation of the h-vector of a non-partitionable complex. By contrast, for a given complex increment it is not always possible to find a complex Gamma such that both Gamma and (Gamma , increment ) are Cohen- Macaulay. We characterize when this is possible, and we show that the construction of such a Gamma in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.