The 36-year-old, who was a free agent after leaving Northampton earlier this month, will remain with the Gigg Lane club until the end of the season.
Platt has admitted that he is looking forward to working with Shakers manager David Flitcroft.
"I'm excited about what I have seen after only one day at the club," he told the club's official website.
"The chairman's vision is something I very much want to be a part of. Obviously I know the manager very well and his enthusiasm rubs off any player, no matter their age.
"What I have seen in and around the club, and from the first training session, has impressed me enormously. I'm excited by this challenge and I look forward to being involved as soon as possible."
Platt scored one goal in 12 appearances for Northampton this season.