  The question whether a partition $\mathcal{P}$ and a hierarchy $\mathcal{H}$
or a tree-like split system $\mathfrak{S}$ are compatible naturally arises in a
wide range of classification problems. In the setting of phylogenetic trees,
one asks whether the sets of $\mathcal{P}$ coincide with leaf sets of connected
components obtained by deleting some edges from the tree $T$ that represents
$\mathcal{H}$ or $\mathfrak{S}$, respectively. More generally, we ask whether a
refinement $T^*$ of $T$ exists such that $T^*$ and $\mathcal{P}$ are
compatible. We report several characterizations for (refinements of)
hierarchies and split systems that are compatible with (sets of) partitions. In
addition, we provide a linear-time algorithm to check whether refinements of
trees and a given partition are compatible. The latter problem becomes
NP-complete but fixed-parameter tractable if a set of partitions is considered
instead of a single partition. We finally explore the close relationship of the
concept of compatibility and so-called Fitch maps.
