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.