Yes you can. But ship water has been carefully and thoroughly filtered, is probably better quality than some "land" water...so why bother? You can always bring an empty bottle and refill it from water coolers on board, if you really need to drink water all day.
But yes, you can lug on bottled water...and sodas..if you must.
I agree, we usually each bring one bottle (maybe 2) and just keep re-filling them. If you are flying in, you have to either ship the water through, costing for perhaps an extra bag or you have to have someone like the taxi driver take you to a store when you are on your way to the ship. TO me this is a bigger bother than it is worth.