Can a 7 Spd Cassette go 6 Spd Bike???

My parts bike has the new 7 spd shifter on it, and my current riding bike has the 6 spd cassette and frame.

So what happens when people put their 7 spd wheel and cassette into the 6 spd frame? I could squeeze it in there, but I'm leery that things will wobble out of their fixed stations. This bike takes MTB abuses and usage, so...