Yep, that's what we do - travel with our refillable stainless steel water bottles for both air and car travel. Way cheaper and more environmental than the disposable (recyclable) plastic bottles. Every airport I've been to has drinking fountains around.
As for drinking fountain water, which some folks turn their noses up at - municipal water is typically much more regulated than bottled water...and because bottled water is so popular, they're likely to not see a lot of use, so I don't worry about germs etc.