  Mixing of active neutrinos with sterile ones generate ``induced''
contributions to the mass matrix of active neutrinos $\sim m_S
\sin^2\theta_{aS}$, where $m_S$ is the Majorana mass of the sterile neutrino
and $\theta_{aS}$ is the active-sterile mixing angle. We study possible effects
of the induced matrix which can modify substantially the implications of
neutrino oscillation results. We have identified the regions of $m_S$ and
$\sin^2\theta_{aS}$ where the induced matrix (i) provides the dominant
structures, (ii) gives the sub-dominant effects and (iii) where its effects can
be neglected. The induced matrix can be responsible for peculiar properties of
the lepton mixing and neutrino mass spectrum, in particular, it can generate
the tri-bimaximal mixing. We update and discuss bounds on the induced masses
from laboratory measurements, astrophysics and cosmology. We find that
substantial impact of the induced matrix is possible if $m_S \sim 0.1-1$ eV and
$\sin^2\theta_{aS} \sim 10^{-3} - 10^{-2}$ or $m_S \geq 200$ MeV and
$\sin^2\theta_{aS} \leq 10^{-9}$. The bounds can be relaxed in cosmological
scenarios with low reheating temperature, if sterile neutrinos decay
sufficiently fast, or their masses change with time.
