As I understand it (ie superficially) provided that your elements are "conforming" then the calculated nodal displacements will converge monotonically on the theoretical value from below, as the mesh density increases. Obviously, this statement applies only to nodes that had not moved during the refinement process.
Crudely, you can consider that in discretising the structure you have imposed additional restraints by requiring the displaced shape of each element to follow the shape function for that element. In effect, you are artificially stiffening your structure. As you increase the mesh density this artificial stiffening reduces.
Most elements are conforming. For non-conforming elements, displacement convergence may not be monotonic.
But you asked about stress convergence. This is almost a meaningless question. Why? Because as you refine your mesh your elements change in size, and probably also in shape. The most meaningful place to extract a stress value is at a gauss point, and these will move as the mesh is refined. Hence you cannot compare like with like.
If you choose to keep some nodes in fixed locations as you refine your mesh, and if you allow your element stresses to be extrapololated to the nodes, then you face the question of which element's extrapolation you wish to use. You will find that for the elements on some sides of the node in question the extrapolated stresses will increase with refinement, whilst for the elements on the other sides of the node in question the extrapolated stresses will decrease. All you can say is that if the convergence of the displacements is monotonic then the convergence of the extrapolated stresses will also be monotonic.
