Below is a response we've sent to a number of customers with this question:
For true convergence, the analysis must converge in all load steps. It is quite common that an analysis will not converge in one of the load steps then in the subsequent steps there is convergence. This does NOT mean convergence and the model is in equilibrium. Most likely the model is not in equilibrium. So be careful. If you get the warning dialog when you read in the results into Interpret then the model is most likely not in equilibrium. There are some exceptions to this rule, say, if you use an extremely small tolerance and a lot of iterations. However due to machine precision, the model tolerance cannot meet the prescribed very small tolerance. The model may still be in equilibrium even though there is no convergence.
What you can do is run a single model with reduced tolerance and increased number of iterations. You can reduce the tolerance by an order of magnitude or more, increase the number of iterations to a few thousand, and run the model and look at the displacements. If the displacements do not change, the model is in equilibrium. You should also make sure to look at mesh sensitivity issues by fining up the mesh, and doing the above again. I also use 6 noded triangles in general for all my analyses to improve accuracy.
If the model does not converge, I let it run to completion then look at the displacements and yielded points to see where my unstable zones are in the model. Even though the model is not in equilibrium, the results are quite useful to determine the zones of instability.