The mesh size is a critical issue which is closely related to the accuracy of the finite element model proposed, the mesh size directly affects/changes the accuracy of the numerical analysis results, is important to choose the best mesh strategy in your finite element model. Also, it is important to take into account that different accuracy can be obtained by different element sizes or types. This question can not be answered by a cualitiative point of view, it is necessary to perform a cuantitative study; like a parametrical study in order to compare the mutual relations between your FE types/sizes, this will provide you the correct mesh size.
Anush, it is not true in general. The convergence also depends on the quality of the mesh (element distortion, grow ratio, etc.). Moreover when a mesh is pretty fine an increment of mesh size does not give great benefit.
The above it's right. You must evaluate what is the right mesh size to analyze your problem, from this size to a lower mesh size your results will almost be equal, and increasingly will be required more computational time for analysis. You must have a balance between the mesh quality and the size of it. A good mesh quality guarantees a more accurate solution of the problem, meanwhile a poor mesh quality will directly affect the possible convergence of the solution and the rate of convergence. Also is important to define if the mesh will be: structured (regular connectivity), unstructured (irregular connectivity) or hybrid (mixture of the above two). Take into account the criteria to evaluate the quality of the mesh (aspect ratio, skewness, orthogonality, smoothness, etc... ). Regards.