  The specific angular momentum of a Kerr black hole must not be larger than
its mass. The observational confirmation of this bound which we call a Kerr
bound directly suggests the existence of a black hole. In order to investigate
observational testability of this bound by using the X-ray energy spectrum of
black hole candidates, we calculate energy spectra for a super-spinning object
(or a naked singularity) which is described by a Kerr metric but whose specific
angular momentum is larger than its mass, and then compare the spectra of this
object with those of a black hole. We assume an optically thick and
geometrically thin disc around the super-spinning object and calculate its
thermal energy spectrum seen by a distant observer by solving general
relativistic radiative transfer equations including usual special and general
relativistic effects such as Doppler boosting, gravitational redshift, light
bending and frame-dragging. Surprisingly, for a given black hole, we can always
find its super-spinning counterpart with its spin $a_*$ in the range
$5/3<a_*<8\sqrt{6}/3$ whose observed spectrum is very similar to and
practically indistinguishable from that of the black hole. As a result, we
conclude that to confirm the Kerr bound we need more than the X-ray thermal
spectrum of the black hole candidates.
