The 24-year-old makeshift right-back suffered the problem during Tuesday night's 1-0 defeat to York in the Johnstone's Paint Trophy and will now be out until October.
Boss Steve Evans said: "He will be out for four weeks. It's disappointing because he's been great for us.
"He's a great professional and we'll miss him. He can play lot of positions and that's why we kept him."
Bradley's injury has been off-set by the return of regular right-backs Nicky Hunt (hamstring) and Dale Tonge (Achilles), who both had a run-out for the reserves on Wednesday.