  We derive general bounds on the Type-III Seesaw parameters from a global fit
to flavor and electroweak precision data. We explore and compare three Type-III
Seesaw realizations: a general scenario, where an arbitrary number of heavy
triplets is integrated out without any further assumption, and the more
constrained cases in which only 3 or 2 (minimal scenario) additional heavy
states are included. The latter assumption implies rather non-trivial
correlations in the Yukawa flavor structure of the model so as to reproduce the
neutrino masses and mixings as measured in neutrino oscillations experiments
and thus qualitative differences can be found with the more general scenario.
In particular, we find that, while the bounds on most elements of the dimension
6 operator coefficients are of order $10^{-4}$ for the general and 3-triplet
cases, the 2-triplet scenario is more strongly constrained with bounds between
$10^{-5}$ and $10^{-7}$ for the different flavours. We also discuss how these
correlations affect the present CMS constraints on the Type-III Seesaw in the
minimal 2-triplet scenario.
