All the cardinals $\kappa\leq\aleph_0$ have the property that there are precicely $\kappa$ cardinals less than $\kappa$. Of course, $\aleph_1$ lacks this property since there are only $\aleph_0 +1= \aleph_0$ cardinals less than it. And I suppose the same goes for all successor cardinals for the same reason.
But are there any limit cardinals for which the property returns? If so, are they in ZFC, or does one have to introduce large cardinals axioms to get them?