Deep neural networks are powerful tools for solving nonlinear problems in science and engineering, but training highly accurate models becomes challenging as problem complexity increases. Non-convex optimization and sensitivity to hyperparameters make consistent performance improvement difficult, and traditional approaches prioritize minimizing mean squared error while overlooking the $L^{\infty}$ norm error that is critical in safety-sensitive applications. To address these challenges, we present HiPreNets, a progressive framework for training high-precision neural networks through sequential residual refinements. Starting from an initial network, each stage trains a refinement network on the normalized residuals of the ensemble so far, systematically reducing both average and worst-case error. A key theme throughout the framework is concentrating training effort on high-error regions of the input domain, which we pursue through complementary techniques including loss function design, adaptive data sampling, localized patching, and boundary-aware training. We validate the framework on benchmark regression problems from the Feynman dataset, where it consistently outperforms standard fully connected networks and reported Kolmogorov-Arnold Networks results, with accuracy approaching machine precision depending on select problems. We further apply the framework to learning the flow map of a 20-dimensional power system ODE, which appears to be the highest dimensional problem studied using this class of multistage methods, achieving substantial reductions in both RMSE and $L^{\infty}$ norm error while enabling a surrogate that predicts system state $238\times$ faster than direct numerical simulation.