Cutting up your credit card into tiny pieces does help prevent the theft of the information especially if you throw away a bit of the credit card into one trash bag and the other into another trash bag.
However when you do cut up your credit card you should cancel the credit card anyway so that it can no longer be used.
So even if someone were to put back all the pieces to your credit card then the information could not be used to make purchases or steal any money.
Still cutting up or shredding your credit card is a good idea to help prevent identity theft.
It's unlikely that someone would come across your credit card cut up and put it back together or at least try to put it back together but anything can happen.
So I always just use my shredder that has a credit card shredder and shred up any expired credit cards to prevent identity theft.