Logit and probit link functions aren't additively separable. So, fitting a model using these link functions implies that the effect of one predictor in determining the outcome is not independent of the values of the other predictors. This might make sense in a lot of contexts, but might be suspect in others.

Take a regression of going to college (0,1) on income and gender, and suppose for a moment that the effect of income is the same for both men and women, but that men are more likely than women to go to college. In a logit (or probit) model, the marginal effect of income will be different for a man versus a woman, because of a negative coefficient on the female dummy -- the model is biased. If you included an interaction, it would seem like the effect of income was weaker for women than me, because the model would be overcompensating for the mis-specified link function. This would be wrong, and misleading.

Of course, linear probability models are wrong for different reasons.

So, what sort of options exist for dealing with mis-specified link functions?