  Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma
transverse to the Seifert fibration. We prove that O is the limit of hyperbolic
cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of
Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the
deformation space constructed from the variety of representations of O-Sigma.
As a corollary of this, we find examples of spherical cone manifolds with
singular set a knot that are not locally rigid. Those examples have large cone
angles.
