There are theories that are non-renormalizable according to standard perturbation theory but are (at the usual level of rigor in theoretical physics) renormalizable when perturbed around the large N limit. See, e.g.,
G. Parisi, The theory of non-renormalizable interactions: The large N expansion, Nuclear Physics B 100.2 (1975): 368-388.
A fully rigorous construction of such a theory has been given in 2 space-time dimensions. See
K. Gawedski and A. Kupiainen, Renormalization of a non-renormalizable quantum field theory, Nuclear Physics B 262 (1985), 33-48.
In general, the only difference between renormalizable and non-renormalizable theories is that the collection of the former/latter forms a finite/infinite-dimensional manifold of theories. But of course, infinite-dimensional manifolds contain lots of finite-dimensional ones; for example those that set all but finitely many of the parameters to zero. The problem is just identifying the right ones to describe given physics is much harder, especially if one aims at an infinitely accurate (''fundamental'') description of Nature.