Let's consider the evidence. Here are six of their best.
Bolton Wanderers 3-0 Swansea Town
FA Cup semi final, Saturday 27 March, 1926
Swansea's first flirtation with the big time was a harsh lesson from one of the powerhouses of early 20th century football in England.
Exeter, Watford, Blackpool, Stoke City, Millwall and Arsenal had been seen-off en route to the semi-final clash with Bolton at White Hart Lane.
But pre-match confidence was wiped out inside 23 minutes as Wanderers swept into a 3-0 lead with goals by Baggett and Smith (2).
Bolton went on to beat Manchester City in the final and didn't meet Swansea again in the tournament until 2012 - when Wanderers won again.
Preston North End 2-1 Swansea Town
FA Cup semi final, Villa Park, 14 March, 1964
Until this season, the closest the Swans had ever got to a Wembley cup final was when they led 1-0 at half-time against Preston in a match evocatively
captured by Pathe News.
Northern Ireland striker Jim McLaughlin's turn and left-foot shot put Swansea in control against a team they had beaten 5-1 earlier in the season.
The match turned on a penalty decision, with Alex Dawson fouled and then scoring from the spot. Swansea hearts were broken when Preston centre-half Tony Singleton, nearly 40 yards from goal, swung his left foot at the ball and watched it fly into the top corner.
It would be 49 years before Swansea were this close again.
Preston North End 1-3 Swansea City
Football League Division Two, 2 May 1981
BBC Sport archive: 1981 - Preston North End 1-3 Swansea City
John Toshack had led Swansea from Division Four to the brink of Division One. A
win at Preston
would seal the deal, and Toshack had assembled a team of stars to deliver top-flight football.
Leighton James set the ball rolling, cutting in from the left and curling the ball into the far corner. Tommy Craig made it 2-0 at half-time, before Alex Bruce's low drive ensured some nervous moments before Swansea counter-attacked to clinch the deal.
The goal commentary is a roll call of legendary Swans: Alan Curtis, to Robbie James, to Jeremy Charles whose left-foot drive meant top-flight football for the first time.
If you want to mark a starting point for the resurgence of Swansea football
this is probably it.
The Supporters' Trust had established a big influence off the pitch and brought in Brian Flynn as manager midway through the season.
From rock bottom in January, the Swans had fought their way into a position where a win would save them from relegation to the Conference.
Hull led 2-1 before James Thomas scored his second penalty of the match to square things up at the interval.
Lenny Johnrose put Swansea ahead before Thomas completed his hat-trick with a cool chip from 20 yards.
Leon Britton was playing, as was future manager Roberto Martinez. Joe Allen was watching from the stands. Against all odds, the future looked bright.
They say it's the richest game in football and, 30 years on from Swansea's first promotion to the top flight, football had moved into the next century - literally and figuratively.
In place of sand-strewn Deepdale there were the carefully-manicured acres of Wembley Stadium and a game to remember.
Swansea led 3-0 at half-time after Scott Sinclair struck twice in two minutes and Stephen Dobie made it three. But thoughts of a procession into the Premier League were wiped out as Joe Allen headed into his own goal and Matt Mills headed in a corner.
Sinclair restored the two-goal advantage with his second penalty to claim a place among English football's elite.
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