A common phrase in U.S. colloquial English is "to play possum." It means "to pretend to be dead." "Possum" is somewhat informal and regional U.S. English for the opossum. Opossums are about the size of an overweight cat and are common throughout the United States. When threatened, opossums really do protect themselves by a physiological response like fainting that makes them look dead. Wikipedia has a picture: http://tinyurl.com/zfed5tv

I'm a native U.S. speaker and I have never heard it. Never. If I heard it I would not understand it. If I read it in a book, I would have to look it up. Where on earth did you find it? According to https://en.wiktionary.org/wiki/sham_Abraham it is archaic slang from the 1700s. It's not found in a U.S. slang dictionary. A search of Project Gutenberg and a regular Google search show, almost entirely, dictionaries and definitions, not real usage. One book gives an explanation. The book dates from 1884, so even in 1884 it was out of date! Abraham Men. This was the name bestowed upon a class of vagabonds who wandered over the country dressed in grotesque fashion, pretending to be mad and working upon the fears or the charity of people for alms. They were common in the time of Shakespeare, and were found even as late as the Restoration. The slang phrase "to sham Abraham," is a survival of the practice. There was a ward in Bethlehem (or Bedlam) Hospital, called the Abraham Ward, and hence probably arose the name of these beggars. Harmless lunatics who had been discharged were often to be seen roaming about the country and were allowed a great deal of licence in consequence of their weak-mindedness. Accordingly, the impostors above mentioned, who used generally to eke out the gifts of the charitable by stealing, when detected in their theft, would plead, as a rule, lunacy as an excuse of their crime.

Never heard of it! I agree with SuKi and Dan - it's NOT common at all! The word SHAM is common enough. But not the expression 'sham Abraham.'

The noun and verb 'sham' are reasonably common. However, like Dan Smith, I had never heard of the expression 'sham Abraham' before today, and could only find it in a dictionary under the heading 'archaic'. So if two fairly well-read and well-educated native speakers of a certain age - such as Mr Smith and myself - have never come across this expression, think it's safe to say that the expression probably doesn't exist in modern-day English.