Those things the big-name editors at the big New York publishers say about their slush piles seem to be true even of tiny editors like us, with our much smaller slush piles. I stop reading the very second I'm sure I don't want to buy a story, and sometimes that second comes on the first page. In most cases,* cover letters don't matter much: I don't care where or whether an author has published anything else--all I care about is whether I like the story that's been submitted to me. I root for every story to win me over, and it bums me out when a story with some promise falls apart. There really aren't enough hours in the day for personalized letters, even acceptance letters, let alone rejections. And several of the stories we're rejecting are stories we actually like, but that, honest and truly, just don't fit the anthology theme. We can imagine other projects they'd work for, but they don't meet our needs at this time.
If you've been going to conferences and you've stopped bothering to take notes at the editor and agent panels because they all sound the same to you, it turns out there's a reason for that. The slush experience is as universal as the coming of age narrative and the hero's journey. If a lost Joseph Campbell manuscript were discovered, and it were called The Editor with a Thousand Submissions, I wouldn't be a bit surprised.
*Except in the rare case of reprints. If the story under consideration has been previously published elsewhere, it does have a better chance if it's from an author with name recognition. That said, most of the slush so far has been previously unpublished.