  We show that both the interior region $r<M-\sqrt{M^2-a^2}$ of a Kerr black
hole and the $a^2>M^2$ Kerr naked singularity admit unstable solutions of the
Teukolsky equation for any value of the spin weight. For every harmonic number
there is at least one axially symmetric mode that grows exponentially in time
and decays properly in the radial directions. These can be used as Debye
potentials to generate solutions for the scalar, Weyl spinor, Maxwell and
linearized gravity field equations on these backgrounds, satisfying appropriate
spatial boundary conditions and growing exponentially in time, as shown in
detail for the Maxwell case. It is suggested that the existence of the unstable
modes is related to the so called "time machine" region, where the axial
Killing vector field is time-like, and the Teukolsky equation, restricted to
axially symmetric fields, changes its character from hyperbolic to elliptic.
