  We demonstrate that the presence of a localized state at the corner of an
insulating domain is not always a predictor of a certain non-trivial
higher-order topological invariant, even though they appear to co-exist in the
same Hamiltonian parameter space. Our analysis of $C^n$-symmetric crystalline
insulators and their multi-layer stacks reveals that topological corner states
are not necessarily correlated with other well-established higher-order
boundary observables, such as fractional corner charge or filling anomaly. In a
$C^3$-symmetric breathing Kagome lattice, for example, we show that the bulk
polarization, which successfully predicts the fractional corner anomaly, fails
to be the relevant topological invariant for zero-energy corner states;
instead, these corner states can be exactly explained by the decoration of
topological edges. Also, while the zero-energy corner states in $C^4$-symmetric
topological crystalline insulators have long been conjectured to be the result
of the bulk polarization at quarter-filling, we correct this misconception by
introducing a proper bulk invariant at half-filling and establishing a precise
bulk-corner correspondence. By refining several bulk-corner correspondences in
two-dimensional topological crystalline insulators, our work motivates further
development of rigorous theoretical grounds for associating the existence of
corner states with higher-order topology of host materials.
