15 minute fit interview: Was asked basic questions that focused on my strengths/weaknesses. Asked to provide examples of challenging situations or situations in which I demonstrated qualities relevant to the consulting role (managing difficult personalities, taking on a leadership role even when not formally appointed, etc.) It was important to be prepared specific with examples as my interviewer kept probing for more details. It is important too in the "fit" part of the interview that everything you say has a purpose; i.e. don't spend time talking about something if it does not demonstrate a character trait that you would like the interviewer to be aware of.
30 minute case interview is typical of consulting cases. I had four interviews before getting an offer, and my sense was that the case interview process varies significantly across interviewers. Generally, in the first round expect more junior interviewers (and hence interview cases that are more data-driven); in latter rounds expect more senior interviewers who will be looking for you to draw insight from the analysis of a problem and provide creative solutions to problems that may appear run-of-the-mill at first pass