  In quantum mechanics, the precision achieved in parameter estimation using a
quantum state as a probe is determined by the measurement strategy employed.
The ultimate quantum limit of precision is bounded by a value set by the state
and its dynamics. Theoretical results have revealed that in interference
measurements with two possible outcomes, this limit can be reached under ideal
conditions of perfect visibility and zero losses. However, in practice, this
cannot be achieved, so precision {\it never} reaches the quantum limit. But how
do experimental setups approach precision limits under realistic circumstances?
In this work we provide a general model for precision limits in two-photon
Hong-Ou-Mandel interferometry for non-perfect visibility. We show that the
scaling of precision with visibility depends on the effective area in
time-frequency phase space occupied by the state used as a probe, and we find
that an optimal scaling exists. We demonstrate our results experimentally for
different states in a set-up where the visibility can be controlled and reaches
up to $99.5\%$. In the optimal scenario, a ratio of $0.97$ is observed between
the experimental precision and the quantum limit, establishing a new benchmark
in the field.
