Travel time distributions obtained from advective transport in multiple realizations of realistic discrete fracture network simulations are analyzed using the truncated one-sided stable distribution, which has previously been shown to generalize both the advection-dispersion solution as well as one-sided stable distributions. Using this model, it is shown that the Fickian assumption inherent in the advection-dispersion equation generally fails, despite the first two moments of travel time essentially scaling linearly with distance. It is also observed that the equally probable realizations drawn from the ensemble can produce a wide range of behavior under the current configuration, such that Fickian conditions are almost obtained in some cases for increasing scales. On the basis of a small-scale calibration against particle breakthrough, the model is then shown to successfully predict limiting bounds of transport for a one order of magnitude increase in scale. Correlation in particle velocity is explicitly shown to be significant for scales close to the characteristic Lagrangian segment length. The network configuration is obtained from extensive site characterization data at the Laxemar region in Sweden and represents a block-scale domain of reasonably sparse background fractures.