Motivated by application in wireless networks, cloud computing, data centers etc, Stochastic Processing Networks have been studied in the literature under various asymptotic regimes. In the heavy-traffic regime, the steady state mean queue length is proved to be $O(\frac{1}ε)$ where $ε$ is the heavy-traffic parameter, that goes to zero in the limit. The focus of this paper is on obtaining queue length bounds on prelimit systems, thus establishing the rate of convergence to the heavy traffic. In particular, we study the generalized switch model operating under the MaxWeight algorithm, and we show that the mean queue length of the prelimit system is only $O\left(\log \left(\frac{1}ε\right)\right)$ away from its heavy-traffic limit. We do this even when the so called complete resource pooling (CRP) condition is not satisfied. When the CRP condition is satisfied, in addition, we show that the MaxWeight algorithm is within $O\left(\log \left(\frac{1}ε\right)\right)$ of the optimal. Finally, we obtain similar results in load balancing systems operating under the join the shortest queue routing algorithm.