Put a new 12-32 cassette (7 speed) on my Peugeot 525 - now winter bike - it shouldn't work with the Shimano short cage mech, but popped the wheel back in and turned the cranks, moved through the gears. Smooth as silk. No adjustments required, didn't have to increase chain length either. Moral seems to be - try it and see!
Edit - correction - I have just recounted my teeth! : The only downside could be the big jump I now have between the 21, 26 and 32 sprockets. The cassette I had been using was a more useful 21, 24, 28. I'm missing that 24!