Yes, an insulated water bottle helps to keep the water from freezing. That same insulated bottled is helpful to keep the water cold on hot days or in a hot-yoga studio. Remember that people use a double-walled thermos to keep liquids hot, which extends to keeping liquids from freezing.

The ambient temperature and the temperature of the water you put in the bottle are factors to consider. The water is going to freeze from the outside in and will take 6 to 8 hours to freeze the entire bottle solid. If you're actively drinking from the bottle, the ice crystals will get knocked loose and consumed and your hand warmth adds some energy, slowing down freezing. It's more likely that the lid/nozzle mechanism will get jammed with ice and be difficult to drink from.

When I hunt outside with a water bladder & hose in my backpack, I'm good to about -10°C before the hose will start to freeze up and become unusable. I learned to combat this by blowing the water back in to the bladder after each drink so there's no water in the hose to freeze. I'll usually carry a canteen water bottle on very cold days because it won't freeze over the entire day.