I am analyzing a FE model made of plates in STAAD.Pro. As I go on reducing the size of the plate mesh, my plate stresses seem to get bigger and bigger and I am not able to arrive at a converged solution. Please advise.

Whereas it is correct that smaller mesh size can improve the accuracy of results but at the same time there are some other factors that one should pay attention to. Check for stress concentrations in the FE model and try to take care of these. For example stress concentration can happen underneath concentrated forces at certain locations of the model or it can happen at a support point or at a re-entrant corner. If you continue reducing the plate size for such cases, the stresses are only going to get higher and higher and the solution will never converge. If there are such occurrences, try addressing those by actions like replacing the concentrated load by a pressure load distributed over a certain area or trying to define some dummy rigid members around the support to spread out the stresses or adding fillets to sharp corners etc.

Overall, in regions where there are higher variations of stress, it is advisable to use a finer mesh. As a general guideline, it is good to have an aspect ratio for the elements to be less than 4:1 (1:1 being the ideal). Another guiding factor is the solution time. Whereas a finer mesh provides more accuracy, it also increases the problem size and is more demanding in terms of computational effort which in turn can slow down the analysis. So one has to strike a balance between the two.