  We study the pseudo-Dirac mixing of left and right-handed neutrinos in the
case where the Majorana masses M_L and M_R are small when compared with the
Dirac mass, M_D. The light Majorana masses could be generated by a
non-renormalizable operator reflecting effects of new physics at some high
energy scale. In this context, we obtain a simple model independent closed
bound for M_D. A phenomenologically consistent scenario is achieved with
M_L,M_R ~ 10^{-7} eV and M_D ~ 10^{-5}-10^{-4} eV. This precludes the
possibility of positive mass searches in the planned future experiments like
GENIUS or in tritium decay experiments. If on the other hand, GENIUS does
observe a positive signal for a Majorana mass \geq 10^{-3} eV, then with very
little fine tuning of neutrino parameters, the scale of new physics could be in
the TeV range, but pseudo-Dirac scenario in that case is excluded. We briefly
discuss the constraints from cosmology when a fraction of the dark matter is
composed of nearly degenerate neutrinos.
