Possibly from the phrase neck and crop, in which crop may refer to the backside of a horse.

come a cropper (third-person singular simple present comes a cropper, present participle coming a cropper, simple past came a cropper, past participle come a cropper)

1. (originally) To fall headlong from a horse.
2. (British, idiomatic) To suffer some accident or misfortune; to fail.

   She came a cropper on the stairs and broke her leg.

   • 1922, Katherine Mansfield [pseudonym; Kathleen Mansfield Murry], “At the Bay”, in The Garden Party, London: Constable & Company, page 7 ↗:

     You couldn't help feeling he'd be caught out one day, and then what an almighty cropper he'd come!

   • 1953, Mervyn Peake, Mr Pye, William Heinemann:

     You tried to convey too much and you conveyed nothing. You came a cropper, major.

   • 2003 November 6, Lynne Truss, “Introduction – The Seventh Sense”, in Eats, Shoots & Leaves: The Zero Tolerance Approach to Punctuation, London: Profile Books Ltd, →ISBN, page 15 ↗:

     We had been taught Latin, French and German grammar; but English grammar was something we felt we were expected to infer from our reading – which is doubtless why I came a cropper over “its” and “it’s”.

• go a cropper