We investigate the effects of gravitational waves (GWs) from a simulated population of binary supermassive black holes (SMBHs) on pulsar timing array data sets. We construct a distribution describing the binary SMBH population from an existing semi-analytic galaxy formation model. Using realizations of the binary SMBH population generated from this distribution, we simulate pulsar timing data sets with GW-induced variations. We find that the statistics of these variations do not correspond to an isotropic, stochastic GW background. The "Hellings & Downs" correlations between simulated data sets for different pulsars are recovered on average, though the scatter of the correlation estimates is greater than expected for an isotropic, stochastic GW background. These results are attributable to the fact that just a few GW sources dominate the GW-induced variations in every Fourier frequency bin of a five-year data set. Current constraints on the amplitude of the GW signal from binary SMBHs will be biased. Individual binary systems are likely to be detectable in five-year pulsar timing array data sets where the noise is dominated by GW-induced variations. Searches for GWs in pulsar timing array data therefore need to account for the effects of individual sources of GWs. © 2012. The American Astronomical Society. All rights reserved..