Gregory, 23, will miss the first four games of the 2016 season. The failed tests occurred during the 2015 season, not after, and an appeals process already has been completed.
At last year's NFL scouting combine, Gregory tested positive for marijuana, which put him in the league's drug program. Considered a first-round prospect out of Nebraska, he fell to the second round, where the Cowboys selected him with the 60th overall pick.
The Cowboys and Gregory agreed to have a personal handler with him during the season to help him cope with issues related to anxiety.
Gregory signed a four-year deal worth roughly $3.8 million after the draft, including $1.4 million guaranteed. It is possible Gregory could lose the $370,000 guaranteed in his 2016 base salary as a result of the suspension. Gregory failed three tests during the season in order to be suspended four games. It will cost him roughly $143,000 of his $608,406 base salary.
Gregory played in 12 games as a rookie and did not record a sack. He suffered a high-ankle sprain in the season opener against the New York Giants and missed the next four games. Cowboys coaches credited him with 11 tackles and 16 quarterback pressures to go along with one tackle for loss.
With Greg Hardy, Jeremy Mincey and Jack Crawford scheduled to be free agents, the Cowboys are hoping Gregory can develop like DeMarcus Lawrence, their 2014 second-round pick. After going without a regular-season sack as a rookie, Lawrence led the Cowboys in sacks in 2015 with eight.
Based on the rules of the program, Gregory will be able to work out with the Cowboys through the offseason program, minicamp, training camp and preseason games. He will also be able to work out on his own at the team's facility during the suspension but will not be allowed to practice.