Yes, a nonlinear analysis will converge to produce the maximum stress for that mesh size.
You have to do a series of smaller element sizes around the location of maximum stress to plot the maximum stress vs. element size to know that you have captured the "true" stress.
Note that most finite element models contain stress singularities, and some of those models have the singularity at the point of maximum stress. If that is the case in your model, you will find that there is no convergence of stress with element size, even though each analysis converged on force and displacement during the solution. If that happens, the model needs some more work.
Please reply with some images of your results.