In this paper we analyse the stability of black hole Cauchy horizons arising in a class of two-dimensional dilaton gravity theories that describes `accelerated' black holes. It is shown that, due to the characteristic asymptotic Rindler form of the metric or to the presence of an acceleration horizon, time-dependent gravitational perturbations generated in the external region do not necessarily blow up when propagated along the Cauchy horizon. There exists, in fact, a region of non-zero measure in the space of the parameters characterizing the solutions such that mass inflation is avoided and the spacetime geometry remains regular on this surface. Despite this fact, however, quantum backreaction seems to produce a scalar curvature singularity there.