I know that some people do this, but I just can't imagine doing it. They just seem so big and bulky to cart around -- they'd take up a bunch of space in a backpack, for example. And unlike a bottle of water, you can't close and seal the lid, meaning that you can't carry a half-filled mug in a backpack.
In the parks, you can get free cups of ice water at all counter-service restaurants ... just ask for some.