One type of model that is sensitive to contact penetration is the interference fit problem of a shaft in a hole. Stress in the parts changes a lot with the amount of penetration. An increase in Normal Stiffness of the contact can reduce penetration, but that is not the only adjustment needed to get accurate results in this model. I found that I also had to adjust the Analysis Setting. Under Nonlinear Controls,  override the Program Controlled Displacement Convervgence Tolerance and type in a much smaller value than 0.5%.