The Patriots will also lose a first-round pick in 2016 and a fourth-round pick in 2017 and have been fined $1 million.
The punishment was announced five days after the release of a lengthy report which found that Patriots personnel deliberately deflated footballs before the AFC Championship Game in January and that Brady "was at least generally aware" of the violations.
The 243-page report by league-appointed attorney Ted Wells said it was "more probable than not" that Brady was aware of plans to prepare the footballs to his liking, below the NFL-mandated minimum of 12.5 pounds per square inch.
The report identified two Patriots employees -- officials' locker room attendant Jim McNally and equipment assistant John Jastremski -- as the ones who executed the plan.
Brady repeatedly stated that he did not know about the efforts to deflate the game balls, but Wells' report found those claims "not plausible and contradicted by other evidence."
Brady's agent said last week that the report contains "significant and tragic flaws" and suggested that the NFL cooperated in a "sting operation" with the Indianapolis Colts, who had alerted the league of their suspicions of the Patriots' use of under-inflated footballs.
New England defeated Indianapolis, 45-7, in the AFC title game before beating the Seattle Seahawks in the Super Bowl two weeks later.