The New Orleans Saints officially were eliminated from playoff contention Sunday.

They were still battling back and forth with the Cowboys in Dallas when the Minnesota defeated the Texans in Houston and made it mathematically impossible for New Orleans to do what had been improbable ever since an 0-4 start.

The Saints’ 34-31 overtime victory against the Cowboys, a week after their most complete performance of the season in a 41-0 thrashing of Tampa Bay, only served to make the impending end of the season that much more difficult for them to swallow.

A team that was 13-3 and one defensive stop away from the NFC Championship game last season was picked by many to play in the Super Bowl in its home stadium this season.

This Saints season will indeed end in the Mercedes-Benz Superdome, but it will happen next Sunday when Carolina shows up and two also-rans finish their insignificant seasons.

The sense of what could have been and maybe even should have been is even more palpable as the season as a whole comes into focus.

There was the game against Kansas City in Week 3 when they led 24-6 late in the third quarter and somehow found a way to lose in overtime to a team that has won just once in the three months since beating New Orleans.

There was the game a week later in Green Bay when Garrett Hartley made a 48-yard field goal that would have given the Saints the lead with less than three minutes to play, but David Thomas was called for holding. After that penalty and a subsequent encroachment penalty on the Packers were enforced, Hartley missed and New Orleans was 0-4.

Then there was a 10-point loss to San Francisco in which Drew Brees threw two interceptions that were returned for scores and a 10-point loss at Atlanta in which Brees threw a career-worst five interceptions.

Teams that belong in the playoffs don’t do those things, and the 2012 New Orleans Saints are living proof.

Brees has since shaken off the uncharacteristic rash of turnovers and been outstanding the last two weeks.

The Saints have found the running game that was missing for much of the first half of the season and brought much-needed balance to the offense. Mark Ingram has gone from apparent bust to a player who might well be what New Orleans envisioned when it traded back into the first round to get him two years ago.

First-year coordinator Steve Spagnuolo’s defense, though still likely headed to the record books for all the wrong reasons, has played well enough of late that this team can finish with a 6-3 run if it beats the Panthers on Sunday.

Amid the months-long attention that was grabbed by the bounty scandal and Roger Goodell and Sean Payton’s absence and hearings and appeals, the most frustrating aspect of this season for these Saints is this:

At the time it became official that they won’t be good enough to make the playoffs, they were just starting to look like a team that might have belonged there.