Ehrenfest 1931 gives an argument to the effect that the application of the spin-statistics theorem to composite systems is valid, but only as an approximation and under certain conditions. Unfortunately this article is behind a paywall. The abstract reads as follows:
From Pauli's exclusion principle we derive the rule for the symmetry of the wave functions in the coordinates of the center of gravity of two similar stable clusters of electrons and protons, and justify the assumption that the clusters satisfy the Einstein-Bose or Fermi-Dirac statistics according to whether the number of particles in each cluster is even or odd. The rule is shown to become invalid only when the interaction between the clusters is large enough to disturb their internal motion.
The quaintness of the abstract is enhanced by the mysterious failure to mention neutrons -- this being explained by the fact that the neutron was not discovered experimentally until the following year. Pretty amusing that the American Physical Society still owns the copyright on a paper this old, one of whose authors was probably paid through my grandparents' and great-grandparents' income taxes.
Googling quickly turns up various loose verbal characterizations of this result, without proof. Can anyone supply an outline of how the argument works and possibly a more precise statement of what the result really is?
Some sources seem to refer to this as the Wigner-Ehrenfest-Oppenheimer (WEO) result or Ehrenfest-Oppenheimer-Bethe (EOB) rule. Historically, Feynman seems to have used this result to show that Cooper pairs were bosonic.
Although I've tagged this question as quantum field theory, and the spin-statistics theorem is relativistic, the 1931 date seems to indicate that this would have been a result in nonrelativistic quantum mechanics. This answer by akhmeteli sketches what seems to be a similar relativistic result from a book by Lipkin.
In the limit of low-energy experiments, the EO rule says the usual spin-statistics connection applies, and this seems extremely plausible. If not, then it would be too good to be true: we would be able to probe the composite structure of a system to arbitrarily high energies without having to build particle accelerators.
Bethe 1997 has the following argument:
Consider now tightly bound composite objects like nuclei. It then makes sense to ask for the symmetry of the wavefunction of a system containing many identical objects of the same type, e.g., many He4 nuclei. This symmetry can be deduced by imagining that the interchange of two composites is carried out particle by particle. Each interchange of fermions changes the sign of the wavefunction. Hence the composite will be a fermion if and only if it contains an odd number of fermions[...]
Note the qualifier "tightly bound," which implies a restriction to low-energy experiments. The abstract of Ehrenfest 1931 seems to say that this is a necessary assumption, but it is never used in Bethe and Jackiw's argument, and this suggests that their argument is an oversimplification.
Fujita 2009 recaps the Bethe-Jackiw argument, but then says:
We shall see later that these arguments are incomplete. We note that Feynman used these arguments to deduce that Cooper pairs  (pairons) are bosonic . The symmetry of the many-particle wavefunction and the quantum statistics for elementary particles are one-to-one . [...] But no one-to-one correspondence exists for composites since composites, by construction, have extra degrees of freedom. Wavefunctions and second-quantized operators are important auxiliary quantum variables, but they are not observables in Dirac's sense . We must examine the observable occupation numbers for the study of the quantum statistics of composites. In the present chapter we shall show that EOB's rule applies to the Center-of-Mass (CM) motion of composites.
Unfortunately I only have access to Fujita through a keyhole, so I can't see what the references are or any of the details that they promise in this beginning-of-chapter preview.
Bethe and Jackiw, Intermediate Quantum Mechanics
P. Ehrenfest and J. R. Oppenheimer, "Note on the Statistics of Nuclei," Phys. Rev. 37 (1931) 333, http://link.aps.org/doi/10.1103/PhysRev.37.333 , DOI: 10.1103/PhysRev.37.333
Fujita, Ito, and Godoy, Quantum Theory of Conducting Matter: Superconductivity, 2009