The recently discovered vast thin plane of dwarf satellites orbiting the Andromeda Galaxy (M31) adds to the mystery of the small-scale distribution of the Local Group's galaxy population. Such well-defined planar structures are apparently rare occurrences in cold dark matter cosmological simulations, and we lack a coherent explanation of their formation and existence. In this paper, we explore the long-term survivability of thin planes of dwarfs in galactic haloes, focusing, in particular, on systems mimicking the observed Andromeda distribution. The key results show that, in general, planes of dwarf galaxies are fragile, sensitive to the shape of the dark matter halo and other perturbing effects. In fact, long-lived planes of satellites only exist in polar orbits in spherical dark matter haloes, presenting a challenge to the observed Andromeda plane that is significantly tilted with respect to the optical disc. Our conclusion is that, in the standard cosmological models, planes of satellites are generally short lived, and hence we must be located at a relatively special time in the evolution of the Andromeda Plane, lucky enough to see its coherent pattern.