Contemporary wireless networks are based on a wide range of di fferent technologies providing overlapping coverage. This o ffers users a seamless integration of connectivity by allowing to switch between networks, and opens up a promising area for boosting the performance of wireless networks. Motivated by this, we consider a networking environment in which users are able to select between the available wireless networks to minimize the mean processing times for file downloads in the presence of background traffic. The information available to the user is only the total number of jobs in each network, rather than the per-network numbers of foreground and background jobs. This leads to a complex partial information decision problem on which we focus. We develop and evaluate a Bayesian learning algorithm that optimally splits a traffic stream that minimizes the expected sojourn time. The algorithm learns as the system operates and provides information at each decision and departure epoch. We evaluate the optimality of the partial information algorithm by comparing the performance of the algorithm with the "ideal" performance obtained by solving a Markov decision problem with full state information. To this end, we have conducted extensive experiments both numerically and in a simulation testbed with the full wireless protocol stack. The results show that the Bayesian algorithm has close to optimal performance over a wide range of parameter values.