  A usual assumption in quantum estimation is that the unknown parameter labels
the possible states of the system, while it influences neither the sample space
of outcomes nor the measurement aimed at extracting information on the
parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao
theorem and to introduce the quantum Fisher information as an upper bound to
the Fisher information of any possible measurement. However, there are relevant
estimation problems where this assumption does not hold and an alternative
approach should be developed to find the genuine ultimate bound to precision of
quantum measurements. We investigate physical situations where there is an
intrinsic dependence of the measurement strategy on the parameter and find that
quantum-enhanced measurements may be more precise than previously thought.
