  Many results in the quantum metrology literature use the Cram\'er-Rao bound
and the Fisher information to compare different quantum estimation strategies.
However, there are several assumptions that go into the construction of these
tools, and these limitations are sometimes not taken into account. While a
strategy that utilises this method can considerably simplify the problem and is
valid asymptotically, to have a rigorous and fair comparison we need to adopt a
more general approach. In this work we use a methodology based on Bayesian
inference to understand what happens when the Cram\'er-Rao bound is not valid.
In particular we quantify the impact of these restrictions on the overall
performance of a wide range of schemes including those commonly employed for
the estimation of optical phases. We calculate the number of observations and
the minimum prior knowledge that are needed such that the Cram\'er-Rao bound is
a valid approximation. Since these requirements are state-dependent, the usual
conclusions that can be drawn from the standard methods do not always hold when
the analysis is more carefully performed. These results have important
implications for the analysis of theory and experiments in quantum metrology.
