  We investigate the effects of the propagation of a non-degenerate Brownian
noise through a chain of deterministic differential equations whose
coefficients are rough and satisfy a weak like H{\"o}rmander structure (i.e. a
non-degeneracy condition w.r.t. the components which transmit the noise). In
particular we characterize, through suitable counterexamples , almost sharp
regularity exponents that ensure that weak well posedness holds for the
associated SDE. As a by-product of our approach, we also derive some density
estimates of Krylov type for the weak solutions of the considered SDEs.
