  We investigate a dichotomy property for Hardy--Littlewood maximal operators,
non-centered $M$ and centered $M^c$, that was noticed by Bennett, DeVore and
Sharpley. We illustrate the full spectrum of possible cases related to the
occurrence or not of this property for $M$ and $M^c$ in the context of
non-doubling metric measure spaces $(X, \rho, \mu)$. In addition, if $X =
\mathbb{R}^d$, $d \geq 1$, and $\rho$ is the metric induced by an arbitrary
norm on $\mathbb{R}^d$, then we give the exact characterization (in terms of
$\mu$) of situations in which $M^c$ possesses the dichotomy property provided
that $\mu$ satisfies some very mild assumptions.
