  The core of Heisenberg's heuristic argument for the uncertainty principle,
involving the famous $\gamma$-ray microscope $\textit{Gedankenexperiment}$,
hinges upon the existence of measurements that irreversibly alter the state of
the system on which they are acting, causing an irreducible disturbance on
subsequent measurements. The argument was put forward to justify measurement
incompatibility in quantum theory, namely, the existence of measurements that
cannot be performed jointly$-$a feature that is now understood to be different
from irreversibility of measurement disturbance, though related to it. In this
article, on the one hand, we provide a compelling argument showing that
measurement incompatibility is indeed a sufficient condition for
irreversibility of measurement disturbance; while, on the other hand, we
exhibit a toy theory, termed the minimal classical theory (MCT), that is a
counterexample for the converse implication. This theory is classical, hence it
does not have complementarity nor preparation uncertainty relations, and it is
both Kochen-Specker and generalised noncontextual. However, MCT satisfies not
only irreversibility of measurement disturbance, but also the properties of
no-information without disturbance and no-broadcasting, implying that these
cannot be understood $\textit{per se}$ as signatures of nonclassicality.
