It means to fail; to fall on your face.
It sounds so "non-west coast" that I would never have thought to employ it, but since you brought it up, I checked out the etymology. Three that I found vary somewhat:
From Word Detective:
The original phrase was "neck and crop," describing a fall from a horse where the rider is thrown headlong over the horse’s head. The most common occasion for this sort of extremely unpleasant accident is when the horse stops short of a jump, as in a steeplechase, but the rider keeps going. "Neck and crop" itself refers to the horse’s head, "crop" being another word for "throat."
From Phrase Finder:
Meaning
Fall over or fail at some venture.
Origin
Probably derives from a headlong ‘neck and crop’ fall. A crop is the handle of a whip
From World Wide Words
We use come a cropper now to mean that a person has been struck by some serious misfortune, but it derives from hunting, where it originally meant a heavy fall from a horse. Its first appearance was in 1858, in a late and undistinguished work called Ask Mamma, by that well-known Victorian writer on hunting, R S Surtees, who’s perhaps best known for Jorrock’s Jaunts and Jollities.
The earliest easily traceable source of cropper is the Old Norse word kropp for a swelling or lump on the body. This is closely related to the Old English word for the rounded head or seed body of a plant, from which we get our modern word crop for the produce of a cultivated plant. In the sense of a bodily lump, it was applied first to the crop of a bird but then extended to other bodily protuberances.
This is where things get complicated: the same word travelled from a Germanic ancestor through Vulgar Latin and Old French back into English as croup for the rump of a horse.
At the end of the eighteenth century English developed a phrase neck and crop, with the sense of “completely”. This is first recorded in a poem by Lady Carolina Nairne:
The startish beast took fright, and flop
The mad-brain’d rider tumbled, neck and crop!
Now neck and crop is a rather odd expression, and we’re not sure how it came to be. It could be that crop is a variant of croup, suggesting that a horse that fell neck and crop collapsed all of a heap, with both head and backside hitting the ground together. Or perhaps crop had its then normal meaning, so the expression was an intensified version of neck, perhaps linked to an older expression neck and heels that’s similar to head over heels.
It’s thought that come a cropper derives from neck and crop, with cropper in the role of an agent noun, referring to something done in a neck-and-crop manner, and that the phrase developed from there.
These examples don’t appear to agree on exactly what the ‘crop’ in the phrase refers to.
