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It is an American cruise ship - Americans are always served iced water with dinner - just ask for ship's water, which is free otherwise they will sell you the bottled stuff. Ask for ice in your cabin. There is always water, tea, coffee and lemonade free to drink around the ship in dispensers, so I cannot see the point of bringing it on. If you do though, just slap a luggage label on the pack of water and it will be delivered to your cabin with your luggage.
When you arrive at the diningroom for your evening meal once seated and given your menu's you will be offered water. As far a bringing it on board no problems, it will be cheaper doing it that way than buying it on the ship unless you are stuck that is.
I've just returned from 8 nights on Splendor in the Med. Iced water was always offered free as soon as we were seated at the dining table and there was no issues with bringing bottled water on board.
Note: if you ask for mineral water at the dining table it will be charged at a couple of dollars a bottle.
Tea, coffee, lemonade & iced tea were always available free of charge from dispensers around the ship and in the informal dining areas.
0.5 litre bottles of water were supplied in the cabins but cost $2.35.
If you prefer soft drinks to alcohol then an unlimited soft drink package at $4/day for kids & $6/day for adults may be a good option.