The nonrelativistic Schrodinger field allows spin independent of statistics, so that you can imagine a nonrelativistic Schrodinger scalar field with Fermionic statistics, or a Schrodinger spinor field with Bose statistics. These models are mathematically consistent, but they are not the nonrelativistic limits of any consistent relativistic fields.
But this doesn't mean that these fields with wrong spin/statistics are realized as effective fields at long-distances in any physical system. Is there an argument that starting from relativistic fields that obey spin statistics, which reduce at low velocities to nonrelativistic fields which obey spin statistics, whenever you make a system with long distance rotational invariance (so that the effective spin of particles makes sense), and translational invariance (so that it's a normal field theory with scattering states), then all the long-distance composite effective fields obey spin/statistics? Is there any nontrivial statement which is true which is spin-statistics in the nonrelativistic context?
There are papers by Berry and collaborators which describe how the nonrelativistic spin/statistics is supposedly natural using a peculiar structure that relates rotations are related to exchanges. I didn't buy these arguments at all, because I couldn't see the point of proving spin/statistics in situations where it clearly isn't true. But perhaps there is a nontrivial correct statement.