  We study the role of probe dimension in determining the bounds of precision
and the level of incompatibility in multi-parameter quantum estimation
problems. In particular, we focus on the paradigmatic case of unitary encoding
generated by $\mathfrak{su}(2)$ and compare precision and incompatibility in
the estimation of the same parameters across representations of different
dimensions. For two- and three-parameter unitary models, we prove that if the
dimension of the probe is smaller than the number of parameters, then
simultaneous estimation is not possible (the quantum Fisher matrix is
singular). If the dimension is equal to the number of parameters, estimation is
possible but the model exhibits maximal (asymptotic) incompatibility. However,
for larger dimensions, there is always a state for which the incompatibility
vanishes, and the symmetric Cram\'er-Rao bound is achievable. We also
critically examine the performance of the so-called asymptotic incompatibility
(AI) in characterizing the difference between the Holevo-Cram\'er-Rao bound and
the Symmetric Logarithmic Derivative (SLD) one, showing that the AI measure
alone may fail to adequately quantify this gap. Assessing the determinant of
the Quantum Fisher Information Matrix (QFIM) is crucial for a precise
characterization of the model's nature. Nonetheless, the AI measure still plays
a relevant role since it encapsulates the non-classicality of the model in one
scalar quantity rather than in a matrix form (i.e., the Uhlmann curvature).
