  One key aspect of quantum metrology, measurement incompatibility, is evident
only through the simultaneous estimation of multiple parameters. The symmetric
logarithmic derivative Cram\'er-Rao bound (SLDCRB), gives the attainable
precision, if the optimal measurements for estimating each individual parameter
commute. When the optimal measurements do not commute, the SLDCRB is not
necessarily attainable. In this regard, the Holevo Cram\'er-Rao bound (HCRB)
plays a fundamental role, providing the ultimate attainable precisions when one
allows simultaneous measurements on infinitely many copies of a quantum state.
For practical purposes, the Nagaoka Cram\'er-Rao bound (NCRB) is more relevant,
applying when restricted to measuring quantum states individually. The
interplay between these three bounds dictates how rapidly the ultimate
metrological precisions can be approached through collective measurements on
finite copies of the probe state. We first consider two parameter estimation
and prove that if the HCRB cannot be saturated with a single copy of the probe
state, then it cannot be saturated for any finite number of copies of the probe
state. With this, we show that it is impossible to saturate the HCRB for
several physically motivated problems. For estimating any number of parameters,
we provide necessary and sufficient conditions for the attainability of the
SLDCRB with separable measurements. We further prove that if the SLDCRB cannot
be reached with a single copy of the probe state, it cannot be reached with
collective measurements on any finite number of copies of the probe state.
These results together provide necessary and sufficient conditions for the
attainability of the SLDCRB for any finite number of copies of the probe state.
This solves a significant generalisation of one of the five problems recently
highlighted by [P.Horodecki et al, Phys. Rev. X Quantum 3, 010101 (2022)].
