  Extracting quasinormal modes from compact binary mergers to perform black
hole spectroscopy is one of the fundamental pillars in current and future
strong-gravity tests. Among the most remarkable findings of recent works is
that including a large number of overtones not only reduces the mismatch of the
fitted ringdown but also allows one to extract black hole parameters from a
ringdown analysis that goes well within the nonlinear merger part. At the same
time, it is well understood that several details of the ringdown analysis have
important consequences for the question of whether overtones are present or
not, and subsequently, to what extent one can claim to perform black hole
spectroscopy. To clarify and tackle some aspects of overtone fitting, we
revisit the clearer problem of wave propagation in the scalar Regge-Wheeler and
P\"oschl-Teller potentials. This setup, which is to some extent qualitatively
very similar to the nonlinear merger-ringdown regime, indicates that using even
an approximate model for the overtones yields an improved extraction of the
black hole mass at early ringdown times. We find that the relevant parameter is
the number of included modes rather than using the correct model for the
overtones themselves. These results show that some standard tests for verifying
the physical contribution of an overtone to a waveform can be misleading, and
that even in the linear case it can be difficult to distinguish the presence of
an excited mode from the fitting of non-QNM effects.
