SUGRA & consistent field theories Why must a consistent theory with a rarita schwinger field (i.e. massless gravitino in the spectrum) be supersymmetric? I was reviewing the GSO projection, Spin Structure, etc. & wasn’t able to make out the argument. Is this an actual theorem (like Weinberg-Witten for composite gravitons for instance) or is it just a belief?
 A: Not sure what you mean by "belief". Composite spin 3/2 fields are permeating our world, viz the Δ(1232) baryon. 
Fundamental  Rarita-Schwinger (R-S) fields propagate acausally, as proven in a historic paper by 
Velo  & Zwanzinger, Phys Rev 188 2218  (1969) available online. 
One of the earliest triumphs of supergravity was Das & Freedman,  Nucl Phys B114 (1976) 271-296 demonstrating that local supersymmetry invariance saves the day by ensuring causality.
The propagator grows like $s\sqrt s$ at high energies,  by Mandelstam,
but unphysical scattering exchanges thereof are prevented by its coupling to the conserved supercurrent, of which it is the gauge field. The same supercurrent is not simultaneously Poincare & local susy invariant, evading the W-W theorem analogously to vector gauge fields. But I assume you are not focussed on loopholes to the exclusions of the W-W theorem; physics theorems normally rely on facts on the ground to ensure they are carefully conditioned to prevent deductions of the nonexistence of fish. 
