  Inspired by the classic problem of Boolean function monotonicity testing, we
investigate the testability of other well-studied properties of combinatorial
finite set systems, specifically \emph{intersecting} families and
\emph{union-closed} families. A function $f: \{0,1\}^n \to \{0,1\}$ is
intersecting (respectively, union-closed) if its set of satisfying assignments
corresponds to an intersecting family (respectively, a union-closed family) of
subsets of $[n]$.
  Our main results are that -- in sharp contrast with the property of being a
monotone set system -- the property of being an intersecting set system, and
the property of being a union-closed set system, both turn out to be
information-theoretically difficult to test. We show that:
  $\bullet$ For $\epsilon \geq \Omega(1/\sqrt{n})$, any non-adaptive two-sided
$\epsilon$-tester for intersectingness must make
$2^{\Omega(n^{1/4}/\sqrt{\epsilon})}$ queries. We also give a
$2^{\Omega(\sqrt{n \log(1/\epsilon)})}$-query lower bound for non-adaptive
one-sided $\epsilon$-testers for intersectingness.
  $\bullet$ For $\epsilon \geq 1/2^{\Omega(n^{0.49})}$, any non-adaptive
two-sided $\epsilon$-tester for union-closedness must make
$n^{\Omega(\log(1/\epsilon))}$ queries.
  Thus, neither intersectingness nor union-closedness shares the
$\mathrm{poly}(n,1/\epsilon)$-query non-adaptive testability that is enjoyed by
monotonicity.
  To complement our lower bounds, we also give a simple
$\mathrm{poly}(n^{\sqrt{n\log(1/\epsilon)}},1/\epsilon)$-query, one-sided,
non-adaptive algorithm for $\epsilon$-testing each of these properties
(intersectingness and union-closedness). We thus achieve nearly tight upper and
lower bounds for two-sided testing of intersectingness when $\epsilon =
\Theta(1/\sqrt{n})$, and for one-sided testing of intersectingness when
$\epsilon=\Theta(1).$
