The versatile former Jets free agent, who even punted for the team in last season's AFC wild-card win over the Bengals, was unable to come to terms with New York early in free agency.
Feely felt that coach Rex Ryan wanted him back, but monetary issues prevented that.
"I know that he wanted me on that team," Feely told ESPN.com AFC East blogger Tim Graham. "Whether they believe in Nick Folk that strongly or have a certain dollar amount allotted to that position, I don't know. But the guys that make the decisions financially made the call. I understand that. I accept that. I'll move on.
"I don't take it personally even though I'm disappointed."
Meanwhile, the Neil Rackers' era in Arizona is nearing an end.
Rackers became a free agent after last season, and the Cardinals had said they wanted to re-sign him. But as time wore on, the decision was made to turn to Feely instead.
"Not knowing where things were going with Neil, we had to look at all the available options," Arizona coach Ken Whisenhunt said. "When we looked at Jay he graded out very high but what was particularly impressive was how well he kicked in some brutal conditions. To kick in that stadium and in that conference and to put up the kind of numbers he did is certainly impressive,"
Rackers made 16 of 17 field goals last season. In his seven seasons with Arizona, he made 165 of 195 field goals and ranks second in franchise history, to kicker Jim Bakken, with 699 points. Rackers' best season was 2005, when he made an NFL-record 40 field goals in 42 attempts.
"It's tough," Whisenhunt said. "I was just thinking about it and my first win as Cardinals head coach was against Seattle on a game-winning field goal by Neil. We're all very grateful to Neil for what he's contributed to our team. He's not only been a very productive kicker but also tremendous in the community and he'll be missed. Unfortunately we couldn't work anything out for him and the most important thing for us was to make sure we were covered at that position with a quality and proven kicker."
During the Jets' open locker room availability Thursday, it was clear the team had made a break with Feely. The veteran's nameplate over his locker had been taken down and replaced by that of Folk, who was signed by the Jets on Feb. 23.
Last season, his second with the Jets, Feely made 30 of 36 field goal attempts, including a 55-yarder, and was a perfect 32-for-32 in point-after attempts.
Still, in a year when every veteran re-signing is being scrutinized by the Jets' front office, those numbers weren't enough to allow Feely his spot on the roster. The Jets asked Feely, 33, to take a pay cut despite his strong season and eventually the parties were at an impasse.
"There's no question that I love that organization," Feely told Graham. "It's a great locker room. The guys on that team are tremendous to be around, a very open and welcoming locker room with a lot of young talent. The foundation is in place for them to be good for a long time.
"The relationship between the coaches and players is unique. It's an ideal fit. The coaches have authority and demand the most, but have the respect of the players, who give their best out of mutual admiration.
"Nobody would want to leave that environment."
Prior to joining the Jets, Feely was cut by Miami after a 2007 season in which he made 21 of 23 field goal attempts. He spent two years with the Giants and spent his first four seasons in the league in Atlanta.
Folk was cut by the Cowboys in Week 15 last season after hitting 18 of 28 field goal attempts.
Information from the Associated Press was used in this report