According to NFL media insiders Ian Rapoport and Mike Garafolo, Mays was cut because of another failed drug test that could now cost him eight or more games in 2016.
Explanation for why #Bengals released Taylor Mays: Per @MikeGarafolo & me, he's facing another drug suspension on top of his original 4 gms
— Ian Rapoport (@RapSheet) July 25, 2016
If S Taylor Mays is ready to play after his latest suspension (for at least 8 games total), Cincy hasn't ruled out bringing him back
— Ian Rapoport (@RapSheet) July 25, 2016
As Rapoport alluded to, it's possible the Bengals look to bring Mays back later in the year if they need some emergency depth at linebacker and/or safety, but we won't see him on an NFL field for at least the first two months of the season, it appears.