While current computational capability has led to finite element analysis becoming the predominant means of assessing three-dimensional stress concentrations, there are nonetheless some three-dimensional configurations where the desired level of accuracy of stresses is not realized on the finest mesh used. Here, we offer some simple means of improving the accuracy of finite element stresses for such configurations, and doing so with modest increases in computational effort. These improved stresses are obtained by using an adaptation of Richardson extrapolation on original mesh results, and also on mesh results with a reduced mesh refinement factor. Verification of the improvements is undertaken using the convergence checks and error estimates reported earlier. The approach is applied to nine three-dimensional test problems. Finite element analysis of these test problems leads to eleven stresses on the finest meshes used that could benefit from being improved. The extrapolation procedure in conjunction with the reduced refinement factor improved all eleven stresses. Error estimates confirmed these improvements for all eleven.