The claim you read is probably
Electronic eigenstates of an atom, or of an inversion-symmetric molecule, always have vanishing electronic dipole moment about the atomic nucleus or the molecular inversion centre.
This is because if you invert the system you must be left with exactly the same eigenstates, but the electric dipole, being a vector quantity, must switch sign.
Magnetic dipole moments, on the other hand, can indeed be nonzero. This is because magnetic dipoles are pseudovector quantities and do not change sign upon inversion. Thus an atom like boron, which has five electrons and therefore a single 2p electron and orbital angular momentum $l=1$, can have a nonvanishing magnetic dipole moment (which will be proportional to the angular momentum of the p electron, so it obeys the standard QM restrictions: e.g. it can only have well-defined components along only one axis at a time).
Electron spin is also a pseudovector quantity, and so atoms and molecules can have spin-induced magnetic dipole moments without violating inversion symmetry.