The eigenvalues of the 1-body reduced state of a fermionic pure state satisfies a set of linear inequalitites, called generalized Pauli constraints. Whenever one of these constraints is saturated (pinning), the state can be shown to exhibit a simplified structure, which can be formulated in terms of selection rules in the basis given by the natural orbitals. On the other hand, examples suggest that some interacting few fermion models tend to have ground states with the property that some of these constraints are close to being saturated (quasipinning). We prove that this implies that the state is well approximated by an exactly pinned one, allowing for a simplified approximate description. Joint work with Christian Schilling and Carlos Benavides-Riveros.