How long can a person survive under water?

Yes, in rare cases. Normally, lack of oxygen will cause brain death within about 5 minutes or so, but in cases of "cold water drowning" in which the victim was immersed in extremely cold water, brain damage is slowed considerably.

It is an axiom among rescue crews that "You're not really dead until you're WARM and dead". I believe that the maximum time between "drowning" and a successful revival is around 40 minutes.

If the rescuers perform CPR while warming the victim, there is a chance - perhaps not a good chance, but a chance! - that the victim can recover with little or no brain damage. This has been observed when small children drown in very cold water; their body mass is so low that they retain very little heat, and they cool quickly into almost a state of "suspended animation". There have been several near-miraculous recoveries in which the victim suffered no long-term ill effects.