It depends on the case. If the case has a molded back that the iPad Mini snaps into, probably/possibly not. The Mini 4 is slightly taller (0.13" taller) than previous Mini models. OTH, if it's held in the case by elastic bands at the corners, it will probably work.
I was wondering if a smart cover for iPad Mini 3, the one that attaches to the side by the magnets would fit the iPad Mini 4. I haven't read anything about that.