  We investigate the see-saw mechanism for generally non-fine-tuned $n \times
n$ mass matrices involving both Dirac and Majorana neutrinos. We specifically
show that the number of naturally light neutrinos cannot exceed half of the
dimension of the considered mass matrix. Furthermore, we determine a criterion
for mass matrix textures leading to light Dirac neutrinos with the see-saw
mechanism. Especially, we study $4 \times 4$ and $6 \times 6$ mass matrix
textures and give some examples in order to highlight these types of textures.
Next, we present a model scheme based on non-Abelian and discrete symmetries
fulfilling the above mentioned criterion for light Dirac neutrinos. Finally, we
investigate the connection between symmetries and the invariants of a mass
matrix on a formal level.
