  The separability problem for word languages of a class $\mathcal{C}$ by
languages of a class $\mathcal{S}$ asks, for two given languages $I$ and $E$
from $\mathcal{C}$, whether there exists a language $S$ from $\mathcal{S}$ that
includes $I$ and excludes $E$, that is, $I \subseteq S$ and $S\cap E =
\emptyset$. In this work, we assume some mild closure properties for
$\mathcal{C}$ and study for which such classes separability by a piecewise
testable language (PTL) is decidable. We characterize these classes in terms of
decidability of (two variants of) an unboundedness problem. From this, we
deduce that separability by PTL is decidable for a number of language classes,
such as the context-free languages and languages of labeled vector addition
systems. Furthermore, it follows that separability by PTL is decidable if and
only if one can compute for any language of the class its downward closure wrt.
the scattered substring ordering (i.e., if the set of scattered substrings of
any language of the class is effectively regular).
  The obtained decidability results contrast some undecidability results. In
fact, for all (non-regular) language classes that we present as examples with
decidable separability, it is undecidable whether a given language is a PTL
itself.
  Our characterization involves a result of independent interest, which states
that for any kind of languages $I$ and $E$, non-separability by PTL is
equivalent to the existence of common patterns in $I$ and $E$.
