Weil sums are finite field character sums that are used to count rational points on varieties in arithmetic geometry. They also tell us the nonlinearity of power permutations used by cryptographers and the correlation properties of sequences used in communications networks. We are interested in Weil sums based on complex-valued additive characters of finite fields that are applied to polynomials with only two monomial terms, that is, binomials. Weil proved a bound on the magnitude of these sums with respect to the usual absolute value. In this talk we are interested in bounds using the p-adic valuation, which tells us about the p-divisibility of our Weil sums, where p is the characteristic of the underlying finite field. We prove an upper bound on the p-divisibility of families of Weil sums of interest in information theory. This is joint work with Philippe Langevin of Universite de Toulon and Sangman Lee and Yakov Sapozhnikov of California State University, Northridge.