I just worked on an FEA model which was essentially a flat, very thin plate of plastic with a pressure applied. Initially, I always use small displacement and linear contact (bonded) for debugging a model just to make sure that loads and boundary conditions are correct. This saves time because large deformation effects can introduce much longer solve times. The linear model solved fine.
When I turned on the large displacement option (nlgeom), the model failed to converge even after several cutbacks (bisections). I certainly did not expect to solve the model in one substep, but I thought the default bisection algorithm would find a converged solution. I was using automatic timestep control. I reduced the initial substep until I eventually obtained a solution, but by this point the subsequent substeps were so small that it would take many, many substeps to complete the load step.
I noticed that after the first substep converged, convergence for later substeps required only a few iterations. Therefore, I broke the load step into 2 separate steps. In the first step, the load was reduced to the value that converged earlier, about 1/1000th. In the second step, I specified the full load with a reasonably sized initial substep. The model had no trouble converging, even though the load had dramatically increased between steps. The reason is that the initial stiffness of the plate is bending only, because it was flat. Since it was a very thin plastic component, the bending stiffness is very low. The first step established some membrane stiffness as the plate tries to assume a more spherical shape. Once the load generates some membrane stress and there is membrane stiffness, subsequent predictions of displacement are more accurate.
I have also used this 2 step load strategy for preloading of bolts. Sometimes, not always, the contact resisting the preload has trouble converging with the full preload. Before contact is established, there is no stiffness resisting the preload. Then, the contact overcloses so much that the solver cannot resolve the overclosure efficiently.