The behavior of the Poisson ratio of a material filled with spherical inclusions is studied in the high-concentration limit, using the Mori-Tanaka and the differential effective medium theories. When the inclusions are either much stiffer or much softer than the matrix, both theories predict the existence of a boundary layer near c = 1 in the graph of the Poisson ratio ν as a function of inclusion concentration c. As c increases, ν first approaches some fixed point ν* that depends only on the matrix properties. In a localized region near c = 1, ν then varies rapidly so as to equal the Poisson ratio of the inclusions at c = 1. The results therefore show a qualitative distinction between, for example, the effect of very hard inclusions and infinitely rigid inclusions. The results also illustrate the extent to which the Poisson ratio fails to obey a mixing law of the Voigt or Reuss type, in that the effective Poisson ratio is not bounded between the Poisson ratios of the matrix and inclusion phases.