Generally, an overly coarse mesh results in underpredicted stresses. Typically, displacement results are less sensitive to mesh quality and density than stress results are.
A good way to ensure that your results are accurate is to perform several iterations of your simulation, with a decreasing mesh size. As the mesh is made denser (smaller elements and a higher element count), the calculated stress level increases. Eventually, the stress results level off, and further mesh density increases have a diminishing effect on the results. When the stress change becomes insignificant between two successive mesh iterations (say less than 1% or 2%) you have achieved what is commonly called mesh convergence or results convergence. Displacement results likely converge before stress results do. When you have demonstrated mesh convergence, you may be confident in the accuracy of your results.
When performing a mesh convergence study, it is best to base the stress comparison on a consistent vertex of the original geometry. Since the mesh is different for each iteration, nodal coordinates vary between iterations. The stress comparison is invalid when the stresses being compared occur at two different sets of coordinates. For this reason, it is best not to base convergence on only the maximum stress in the model, regardless of its location.