The macrodispersion model from stochastic transport theory is demonstrated to be of limited utility when applied to heterogeneous aquifer systems containing narrow connected pathways. This is so even when contrasts in hydraulic conductivity (K) are small and variance in ln K is less than 0.10. We evaluated how well an advection-dispersion model (ADM) could be used to represent solute plumes transported through mildly heterogeneous three-dimensional (3-D) systems characterized by a well-connected dendritic network of 10 cm wide high-K channels. Each high-K channel network was generated using an invasion percolation algorithm and consisted of ~10% by volume high-K regions. Contrasts in K between the channels and matrix were varied systematically from 2:1 to 30:1, corresponding to ln K values ranging from 0.04 to 1.05. Simulations involved numerical models with 3-D decimeter discretization, and each model contained 2--4 million active cells. Transport through each channel network considered only the processes of advection and molecular diffusion. In every case, the temporal change in the second spatial moment of concentrations was linear, with R2 values ranging from 0.97 to 0.99. The third spatial moment, or alternatively, the skewness coefficient values, indicated significant tailing downstream of the plume center. For each case, a corresponding ADM was used to simulate transport through the system. The corresponding ADM employed the effective mean hydraulic conductivity that reproduced the total discharge through the channel network system under an identical ambient gradient. Dispersivity values used in the ADM were obtained from the temporal change in the second spatial moments of concentrations for the plumes in the channel network systems and ranged from 0.014 m to 0.85 m. The results indicate that as the conductivity contrast between the channels and matrix increased, the simulated plumes in the channel network system became more and more asymmetric, with little solute dispersed upstream of the plume center and extensive downstream spreading of low concentrations. Distinctly different spreading was found upstream versus downstream of the plume center. The ADM failed to capture this asymmetry. Comparison of each plume in the channel network system with the corresponding plume produced using the corresponding ADM showed a maximum correlation of only 0.64 and a minimum fractional error of 0.29 for cases in which the log K variance was ~0.20 (ln K variance was ~1.0). At early times the correlations were as low as 0.40. The greatest correlation occurred at late times and for cases in which a wide source was considered.