The objective of this work is to determine whether conventional solute macrodispersion theories adequately predict the behavior of individual subsurface plumes over time and length scales relevant to practical risk assessment and remediation activities. This issue is studied with a set of high-resolution numerical simulations of conservative tracers moving through saturated two-dimensional heterogeneous conductivity fields. The simulation experiments are designed to mimic long-term field studies. Spatially correlated statistically stationary lognormal conductivity realizations are generated with a Fourier transform procedure. Steady state velocity solutions are calculated for these fields using an accurate Darcian solver. Velocity contour plots reveal the presence of disconnected networks of preferential pathways over a range of correlation lengths. Reverse flow cells are rare. The velocity probability density functions have exponential tails and strong longitudinal asymmetries. Solute concentrations are derived from the simulated velocity fields with an accurate taut-spline transport code that minimizes numerical dispersion. The resulting plumes are tracked for travel distances of over one hundred spatial correlation lengths, corresponding to scales of practical interest. First moments and macrodispersivities, which measure the location and extent of the plume, are reasonably well approximated by conventional Fickian theories but continue to vary after long travel distances. This temporal variability highlights the slow convergence of the plume moments to Gaussian limits. Calculations of the relative dilution index, which is a measure of the plume mixing state, indicate strongly non-Gaussian behavior. Other measures of plume structure suggested by anomalous dispersion theories also reveal the persistence of non-Gaussian behavior after long travel distances. These measures appear to be more sensitive to non-Gaussian behavior than the spatial moments. Taken together, the simulation results suggest that conservative solute plumes moving through statistically stationary random media may not converge to Gaussian limits even after traveling hundreds of log conductivity correlation scales.