  If massive neutrinos are the Majorana particles and have a normal mass
ordering, the effective mass term $\langle m\rangle^{}_{ee}$ of a neutrinoless
double-beta ($0\nu 2\beta$) decay may suffer significant cancellations among
its three components and thus sink into a decline, resulting in a "well" in the
three-dimensional graph of $|\langle m\rangle^{}_{ee}|$ against the smallest
neutrino mass $m^{}_1$ and the relevant Majorana phase $\rho$. We present a new
and complete analytical understanding of the fine issues inside such a well,
and discover a novel threshold of $|\langle m\rangle^{}_{ee}|$ in terms of the
neutrino masses and flavor mixing angles: $|\langle m\rangle^{}_{ee}|^{}_* =
m^{}_3 \sin^2\theta^{}_{13}$ in connection with $\tan\theta^{}_{12} =
\sqrt{m^{}_1/m^{}_2}$ and $\rho =\pi$. This threshold point, which links the
{\it local} minimum and maximum of $|\langle m\rangle^{}_{ee}|$, can be used to
signify observability or sensitivity of the future $0\nu 2\beta$-decay
experiments. Given current neutrino oscillation data, the possibility of
$|\langle m\rangle^{}_{ee}| < |\langle m\rangle^{}_{ee}|^{}_*$ is found to be
very small.
