This question was motivated, but is separate from, the question I posted here: How can I improve the predictive power of this logistic regression model?.
In that case the 'cancer' outcome was occurring with ~92% probability. It was commented to me that "these variables don't discriminate your data very well. Since most people have cancer in this data set you can do just as well at predicting whether they have cancer by just saying they all have it." In this instance the predictor variables were poorly chosen and it may not have mattered much what proportion of people had cancer.
Thinking more generally, at what point does the preponderance of one outcome become sufficiently great that logistic regression becomes a poor choice? Are there any rules of thumb to guide judgement in this area?