I do it all the time with no problem. Anything to avoid paying $2 for a bottle of water. I just wish the recycling trash cans were more prevalent in the parks (and used more by guests).
We have taken bottled water into the parks before and no one has said anything. Also, remember that you can get a cup of ice water free at any counter service restaurant. I was a little skeptical about this until we tried it. It works like a charm. You don't have to buy anything either. One time I asked for it with my meal and it was free. And to test it, I went up before we left and just ordered 3 glasses of ice water. No problem! A little easier than carrying bottles of water around.
I don't believe there is any official policy stating that no outside food or drinks are allowed. In one of the pamphlets I received when I checked into a Disney hotel there was a Question and Answer section. One of the questions was asking what should you bring into the parks. The answer from Disney was water, snacks, sunscreen, etc.
So, they actually recommended that you bring water
Every year that we go, we bring a backpack and stuff frozen water in the bags. It saved on money plus we used themto cool off on those hot days. This year we went to target bought a backpack that is a cooler. We were able to get 6 cans of soda in or 5 large frozen waters. I also stuck cold diaper wipes to wipe my kids faces off when it go to hot. No one stopped me and I dont think they will either. It saved us a ton of money since my kids are always crying they are thirsty. I also bought little indiviual packets of coolaid and crystal light we poured them into the water and the klids loved it.
and if your room has a freezer, freeze your water bottles and they will stay nice and cold all morning. If you head back to room room around noon you can always get a new ice cold frozen water bottle for the afternoon.
The only food-related prohibitions are no glass (except baby food jars and perfume bottles), no alcohol, and no hard-sided or rolling coolers. And at Animal Kingdom, no straws and no fast-food-type drink lids.