A number of Type 316 stainless-steel right circular cylindrical shells of varying lengths have been analyzed using two-dimensional discrete-ordinates transport methods together with first- and last-flight particle estimators to aid in the design of neutron collimators for the Tokamak Fusion Test Reactor (TFTR). In the TFTR, the 14-MeV neutron source has a very large spatial extent, and the collimators must be designed to allow spectral measurements that refer to only a small spatial region of this extended source. The analysis identifies the 14-MeV neutrons from scattering in the Type 316 stainless steel immediately adjacent to the collimator opening as the dominant contributor to detector background. Collimator lengths >0.60 m were found sufficient to attenuate uncollided background neutrons for reasonable source-detector distances. The lower energy (<13.8 MeV) neutron background and gamma background were not found to be significant.