I wouldn't bring a water bottle, many places provide you with water anyway.
Bring a water bottle, and leave it in the room where interviewees sit during down time. I think it would look tacky to carry it around with you while you are interviewing, but there's no reason you can't drink it during down time.
This is generally viewed as inappropriate, Aquaman.
I distinctly remember last interview season there was this guy that always brought a water bottle with him to interviews. Well, at one program (that I won't name), the PD saw this interviewee with the water bottle and said "who's this mother-fu*king high and mighty MS4 think he is bringing water to a derm interview? Is he out of his fu*king mind bringing water in here? Get this son of a bi*ch out of my sight before I puke in my handbag!!"
So, be careful with this hot potato(e).