A coney-catcher was a thief or con man.
The term was used in 1592 pamphlets by Robert Greene, the titles of which included "The Defence of Conny-catching," in which he argued there were worse crimes to be found among "reputable" people, and "A Disputation betweene a Hee Conny-catcher and a Shee Conny-catcher."
1. Bryson, Bill. Shakespeare: The World as a Stage. HarperPress, 2007
2. "Coney Catching," http://internetshakespeare.uvic.ca/Library/SLT/history/coney-catching.html
3. "Robert Greene's Social Pamphlets," http://www.bartleby.com/214/1604.html