  General relativity's no-hair theorem states that isolated astrophysical black
holes are described by only two numbers: mass and spin. As a consequence, there
are strict relationships between the frequency and damping time of the
different modes of a perturbed Kerr black hole. Testing the no-hair theorem has
been a longstanding goal of gravitational-wave astronomy. The recent detection
of gravitational waves from black hole mergers would seem to make such tests
imminent. We investigate how constraints on black hole ringdown parameters
scale with the loudness of the ringdown signal---subject to the constraint that
the post-merger remnant must be allowed to settle into a perturbative,
Kerr-like state. In particular, we require that---for a given detector---the
gravitational waveform predicted by numerical relativity is indistinguishable
from an exponentially damped sine after time $t^\text{cut}$. By requiring the
post-merger remnant to settle into such a perturbative state, we find that
confidence intervals for ringdown parameters do not necessarily shrink with
louder signals. In at least some cases, more sensitive measurements probe later
times without necessarily providing tighter constraints on ringdown frequencies
and damping times. Preliminary investigations are unable to explain this result
in terms of a numerical relativity artifact.
