Montana, who captured his titles with san Francisco but played his final two seasons with Kansas City, told USA Today that he couldn't understand how the Patriots could let Brady depart after 20 seasons and six titles together.
"I don't know what's going on inside there, but somebody made a mistake," Montana said. "I think when you look at the whole situation, you try to figure out how you want to get away from things that are there.
"I still don't understand how New England let him get away. I don't understand that."
Brady, who turns 43 before the 2020 season kicks off, signed a two-year free agent deal with the Buccaneers after splitting with coach Bill Belichick and the dynasty he helped build.
"It's not about appreciation. He wants control," Montana said. "I mean, he wants a lot of control. I don't know what Tampa Bay gave him, but at some point in time, you're just a player. You can try to get what you can and do what you want, but in the end, you're still not in the hierarchy when it comes to hiring people, firing people and all that.
"I don't know exactly what he's looking for, but my understanding was that he's just looking for more control of the offense. But I don't know."
Montana, 63, spent 13 seasons in San Francisco and directed the 49ers to Super Bowl titles in the 1981, 1984, 1988 and 1989 seasons and was a boyhood idol of Brady.
An elbow injury suffered in the 1991 pre-season kept Montana out for almost two full seasons, allowing Steve Young to cement himself in a starter's role and setting the stage for the 49ers to trade Montana, then 36, to Kansas City in April 1993.
Montana sparked the Chiefs in two playoff runs, finishing one win from the 1994 Super Bowl before retiring in 1995.
Just as Peyton Manning departed Indianapolis late in his career for a second Super Bowl title run with Denver and Montana made his run with the Chiefs, so Brady will try to have an epic closing act with Tampa Bay.
"It actually brings a new excitement to you, to a certain degree, because it's not going to be the same-old, same-old going into the same locker room that you've been going into for so many years, seeing the same people over and over," Montana said of changing teams after a long and successful stay.
"He doesn't need a fresh start, but it gives you a great feeling inside, looking forward to trying to help the team move forward. And everybody believes in him, looking forward to watching him play."