Benj,
Last year I interviewed at Case on the first possible day, which was something like 7/26, I believe.
Numerically they are a difficult school to get into, with only 4% of the applicants gaining admission, but closer to 40% of interviewees gaining admission.
I had a really great time there, but I don't think my experience was normal, so don't base all of your judgements off my experience alone. I say that because I knew from the get-go that I would be accepted.
My interviewer placed me in the clinic after the interview day was finished (around noon), and I worked alongside D3 and D4 students for 4-5 hours. The interviewer even let me wear his white coat in the clinic!
After that, he drove around town for a tour, stopped and bought us chinese food, and then dropped me off at the airport so I wouldn't have to take the shuttle.
From all of this, I sensed I was in. Now, having said that, all the students who I worked with in the clinic said that anybody who interviews in the first 2 weeks of possible dates at Case usually gets in.
Based on this, I'd say your chances are very good.
Enjoy yourself, and congrats!