On the album "Rivers of Delight" by Vermont's Word of Mouth Chorus (a wonderful recording, by the way), there is one song that uses the style Pinetop Slim mentioned. I can't recall the title, and my Original Sacred Harp book is still packed away after my recent move so I can't look it up, but it's the song that has a verse beginning, "When I was sinking down, sinking down, sinking down..." On the last line of each verse, one of the tenors comes in very strongly about a half-beat early.
I've also heard it in John Roberts and Tony Barrand's recording of "Babylon is Fallen" (not in the Original Sacred Harp, but I think it may be in Southern Harmony) where one of them starts "Ba...bylon is fallen" a full beat earlier than the other, and holds that first syllable for the extra beat.
I like the effect. I imagine it started as a way to keep the other singers moving, and then some people liked the sound and "formalized" it.