The Gunners, who finished fourth in the Premier League last term, claimed their first trophy since 2005 by winning the FA Cup last month.
Despite their cup success, Adams believes that Arsene Wenger's side will struggle to improve their league position next year.
"The club came under criticism for not winning a trophy and I was one of them [who criticised the team]. It was important for them to win that," he told the Daily Express.
"Hopefully it will be a pivotal moment in the careers of the squad members and they can go on and win league titles. However, I don't think they will win the title as Chelsea and Manchester City are blowing the league apart and Manchester United will come again.
"I don't think Arsenal will finish any higher than fourth at the moment, which is a little bit sad, but they needed to win the FA Cup because they had stick from the fans."
Arsenal beat Hull City 3-2 in the FA Cup final at Wembley.