The Highbury legend won four Premier League crowns with the north London club but believes manager Arsene Wenger cannot win trophies due to first-team departures
The north London club last won a trophy in 2005 when they picked up the FA Cup after a penalty shoot-out, but since then have finished no higher than third place in the league.
The former centre-back believes the club must aim higher and blamed player sales for the lack of success.
He told BBC Radio 5: "It's about winning, and how do you win? You have the best players, and if you keep selling your best players you're not going to win.
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"If you sell your best players you're not going to challenge for titles. You need to keep your best players and in recent years we've let them go a bit easy.
"You don't let them go into their last year and run their contract down. I never got anywhere near the last year of my contract."
The north London club lost their third Premier League match of the season away at Manchester United on Saturday, with former striker Robin van Persie opening the scoring for the hosts.
This summer's departure of the 2011-12 Premier League top scorer followed the summer sales of Samir Nasri and Gael Clichy the year before to champions Manchester City, as well as that of Cesc Fabregas to Barcelona.
The disruption late in last season’s transfer window caused a sharp dip in form for the club, but the side managed to recover to qualify for the Champions League in third place, with Van Persie scoring 30 league goals.
Adams believed this was the limit of the club's ambition, and went on to express his support for manager Arsene Wenger, adding: "If Wenger gets them into the top four he's done brilliantly."