  We prove that the tautological rings
$\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})$ and
$\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq 2$
and $2g+n\geq 24$, extending results of Petersen and Tommasi in genus $2$. The
proof uses the intersection of tautological classes with non-tautological
bielliptic cycles. We conjecture the converse: the tautological rings should be
Gorenstein when $g=0,1$ or $g\geq 2$ and $2g+n<24$. The conjecture is known for
$g=0,1$ by work of Keel and Petersen, and we prove several new cases of this
conjecture for $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ when $g\geq 2$.
