Success!
I took one magnet of a type similar to Joe's and waved it back and forth (lontitudally) over a magnetised screwdriver keeping a small gap with a piece of plastic. The result: the screwdriver was partially, but not fully, demagnetised.
Repeated the same process but gradually increasing the distance between the magnet and the screwdriver and it was de-magnetised (almost) fully. 
So, the key to using these devices is to repeat the pass through at gradually increasing distance. I guess that's what the steps seen on some of them are for. Not much use for the OP's digital calipers though!
Halbach array — interesting, it might work better.
Russell.