Yet for as far as they’ve come — not only this season but the last several — it’s hard for this story not to feel a little warm and fuzzy: The team that practices in the gym where they shot “Hoosiers’’ is one win from playing in the Final Four. In downtown Indianapolis, no less, only a few miles from home.

Coming off a win over Syracuse, the fifth-seeded Bulldogs (31-4) put their 23-game winning streak on the line against No. 2 Kansas State today in the West Regional final. Winner moves on. Loser goes home. For Butler, of course, the destination will be the same either way.

“When we sat down and made goals and thought about the season, we said, ‘Why not shoot for the ultimate goal?’ ’’ forward Matt Howard said.

Indeed, why not?

Though this is their first trip this far in the NCAA Tournament, the Bulldogs have been on the national stage for more than a decade now.

They are quick, can shoot, and play good defense, as they proved in their 63-59 win over top-seeded Syracuse. The prevailing wisdom before that game was that the team that shoots 34 percent from 3-point range would have to be even better — in every way — to stay on the court with the Big East regular-season champions. Fact is, the Bulldogs weren’t all that wonderful, and didn’t have to be.

They went 6 for 24 from beyond the arc and, after a nicely played first half, struggled in the second and even fell behind by 4 points late. They answered with an 11-0 run to take control — highlighted by Willie Veasley’s 3-pointer from the corner that circled the rim, caromed high off the backboard, then finally dropped clean through the net.

Count Kansas State coach Frank Martin among the believers.

“Whether your name is Butler, UCLA, Kentucky, whoever, if you’re playing in the Elite Eight game, you’re a very good basketball team,’’ he said. “You shouldn’t be concerned about appearance. You should be concerned about the team.’’

Regardless of the opponent, the Wildcats (29-7) would be in for a challenge, needing to gear back up after a double-overtime 101-96 victory over Xavier in what is already going down as one of the best games in tournament history.

Almost to a man, the K-State players admitted they got very little sleep after the game.

“I just laid there staring at the wall,’’ said Jacob Pullen, whose two 3-pointers and two free throws helped put the game away in the second OT.

The goal for both programs, however, is not to treat this like it’s the only chance they’ll get. It used to be easier to believe at a place like Kansas State, which plays in the shadow of Kansas, but is still in the Big 12 and has the money and facilities to chase the best players.

Scoff at the mid-major label as much as it wants, Butler will always have some mid-major issues to deal with: less money, less national TV exposure, less access to the best players.

“You’ve got to find your niche, and find the right guys to fit your niche,’’ coach Brad Stevens said. “As you can see, mid-major, major, low major, whatever the case may be, we’ve got pretty darn good basketball players here.’’