  Under the assumption that astrophysical black hole candidates are the Kerr
black holes of general relativity, the continuum-fitting method and the
analysis of the K$\alpha$ iron line are today the only available techniques
capable of providing a relatively reliable estimate of the spin parameter of
these objects. If we relax the Kerr black hole hypothesis and we try to test
the nature of black hole candidates, we find that there is a strong correlation
between the measurement of the spin and possible deviations from the Kerr
solution. The properties of the radiation emitted in a Kerr spacetime with spin
parameter $a_*$ are indeed very similar, and practically indistinguishable,
from the ones of the radiation emitted around a non-Kerr object with different
spin. In this paper, I address the question whether measuring the Kerr spin
with both the continuum-fitting method and the K$\alpha$ iron line analysis of
the same object can be used to claim the Kerr nature of the black hole
candidate in the case of consistent results. In this work, I consider two
non-Kerr metrics and it seems that the answer does depend on the specific
background. The two techniques may either provide a very similar result (the
case of the Bardeen metric) or show a discrepancy (Johannsen-Psaltis
background).
