Improved astrophysical reaction rates for {sup 116,120}Sn(n, [gamma]) are of interest because nucleosynthesis models have not been able to reproduce the observed abundances in this mass region. For example, previous s-process calculations have consistently underproduced the s-only isotope 116Sn. Also, these studies have resulted in residual reprocess abundances for the tin isotopes which are systematically larger than predicted by reprocess calculations. It has been suggested that these problems could be solved by reducing the solar tin abundance by 10-20%, but there is no experimental evidence to justify this renormalization. Instead, it is possible that the problem lies in the (n, T) cross sections used in the reaction network calculations or in the s-process models. One reason to suspect the (n, [gamma]) data is that previous measurements did not extend to low enough energies to determine accurately the Maxwellian-averaged capture cross sections at the low temperatures (kT=6-8 keV) favored by the most recent stellar models of the s process. Also, the two most recent high-precision measurements of the 12°Sn(n, [gamma]) cross section are in serious disagreement. Because of its small size, this cross section could affect (via the s-process branching at 121Sn) the relative abundances of the three s-only isotopes of Te.