Lately, there has been a renewed interest in fermionic one-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove that this polytope's volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the one-body space grows and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue that the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.