Inspired by this question (Is there any proof that Jaime Lannister is actually a good swordsman?), the OP mentions Ned in a way that he thinks Ned is a great swordsman. I don't recall anything from the books or show that indicate anything at all about his swordsmanship, he's usually talked about it terms of his honor, or skill as a commander. Aside from vague "we were great once, Ned" comments from Robert (and weren't we all great in our youth? :) ), what evidence do we have that Ned was notable as a swordsman?
No there is no such proof or indication about Ned's talents.
He was an aristocrat so obviously he received martial training but never quite shone like many other big names of Westeros.
As you noted, he is known for his honesty and honor, not for his skills as a fighter.
He must have been good enough however as he fought directly in many battles including the raid at Tower of Joy but there is no evidence suggesting that he was a great warrior.
Others like Jaime, Arthur Dayne, Barristan Selmy, Loras Tyrell, Garlan Tyrell, Oberyn Martell, Lyn Corbray, Qhorin Halfhand etc are all described as exceptionally gifted. Even Robert was counted as a great warrior once. The same has never been said about Ned.
And he is not alone in it, the same has never been said about Tywin Lannister, Jon Arryn or Hoster Tully either. Mace Tyrell is explicitly called out as an incompetent Commander. I am actually glad that ASOIAF doesn't follow the silly tropes that the Good Guy (there are no good guys in the series but that's how some people view Ned) is supposed to be a great warrior, oh and warning, TV Tropes link. Ned has other redeeming talents, he was great at administration and laws.
So in conclusion, there is neither any evidence, nor a claim in the first place that Eddard was among the biggest fighting stars of Westeros.
And I have found a Citadel entry on this same question. This is what GRRM is reported to have said himself:
GRRM has previously stated that Eddard was a competent swordsman. However, Brandon was the real swordsman in the family (SSC).
If George says he was a competent (and therefore not an exceptional one) swordsman, I guess that settles it. Good enough but not Excellent.
In the episode, The Wolf and the Lion, following conversation took place:
BARRISTAN: Life is strange. Not so many years ago we fought as enemies at the Trident.
NED: I'm glad we never met on the field, Ser Barristan ... as is my wife. I don't think the widow's life would suit her.
BARRISTAN: (chuckles) You're too modest. I've seen you cut down a dozen great knights.
NED: My father once told me you were the best he'd ever seen. I never knew the man to be wrong about matters of combat.
I think this speaks to Ned's skill as a swordsman. Also, dozens, perhaps hundreds of Knights and thousands of soldiers died during Robert's rebellion - the definitely were not all great, as evidenced by their deaths and Ned Stark's survival.
In a roundabout way we know that Ned was not as good a swordsman as his brother Brandon was. From George himself:
[Correspondent] I was wondering if you would comment on Benjen Stark's fighting ability. Is he on a level with Brandon, or is he more like Ned?
[GRRM] Depends on the kind of fight you had in mind.
[GRRM] Brandon was the best of the Starks with sword in hand, and the best jouster as well. But Benjen has other skills that serve him well as a ranger... and Ned was likely the best battle commander.
So Spake Martin, About Benjen Stark, December 15, 2000
Now George was comparing the three brothers, and just because Brandon was better doesn't mean Ned was a slouch, Brandon could have been second to none, but I think that if Ned wasn't even the best in family, he would not have been considered an elite swordsman in Westeros. As shown, Ned's talents lay elsewhere, in command.
In the books, Ned's only a competent swordsman, maybe somewhat above average since he survived many important battles. His brother Brandon was a really good swordsman, though Ned was rumored to have killed Ser Arthur Dayne by himself. (The show clarifies that this isn't quite how it went down.)
In the show, Brandon's skills were never mentioned and Ned was shown to be a highly skilled swordsman who killed Ser Gerold Hightower pretty much by himself and lasted quite some time even against the fabled Ser Arthur Dayne. Even later, when he was past his prime, he was able to duel Ser Jaime Lannister fairly evenly for a while.
If you've actually gone into combat, actually faced and crossed men with your sword, and have come out of it with all your limbs intact - you're a pretty good swordsman. Better than most probably.
This is AFAIK the case for Eddard Stark in the books as well (Battle of the Bells at least, right?), and that's ignoring the initial advantage of being trained in sword-fighting, which is a privilege few people have - and itself makes you pretty decent in relative terms (unless you totally bungle it up like Robin Arryn in the show).