  In this work we investigate if the interaction of the Kerr-Taub-NUT spacetime
with test scalar and neutrino fields can lead to the destruction of the event
horizon. It turns out that both extremal and nearly extremal black holes can be
destroyed by scalar and neutrino fields if the initial angular momentum of the
spacetime is sufficiently large relative to its mass and NUT charge. This is
the first example in which a classical field satisfying the null energy
condition can actually destroy an extremal black hole. For scalar fields, the
modes that can lead to the destruction of the horizon are restricted to a
narrow range due to superradiance. Since superradiance does not occur for
neutrino fields, the destruction of the horizon by neutrino fields is generic,
and it cannot be fixed by backreaction effects. We also show that the extremal
black holes that can be destroyed by scalar fields correspond to naked
singularities in the Kerr limit, in accord with the previous results which
imply that extremal Kerr black holes cannot be destroyed by scalar test fields.
