Because they don't have enough time to go grab their own water bottle during time outs, so a "water boy" runs around and ensures all who are thirsty get something to drink. They used to let them hold their own bottles, but a lot of players would "mouth" the nozzles of the bottles instead of squirting it into their mouths without touching their lips. This was unhygienic and would quickly spread sickness amongst the players on the team, therefore they now have a "water boy" hold the bottle and squirt it in their mouths, to ensure that no one touches it and shares their saliva with any other players.
Edit by smorin1487: this may be partially true, but the real reason is that NFL players wear very sticky gloves, and grabbing the water bottles (which will often be wet due to condensation or from being dunked in water coolers to fill up faster) will be wet, and this will affect the stickiness of the gloves.
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