Anytime you begin to lose stiffness in a static structural analysis, there is always potential for convergence trouble.
The sensitivity depends on the application. How localized is the material yielding? How well is the structure constrained?
Perhaps the structure is beginning to buckle and non-convergence is a reflection of a physical instability.
You could test this theory by post processing your force-deflection curve to determine if you have lost all resistance to further loading.
You could also try adding stabilization damping to make convergence more forgiving to material softening.
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