Does anyone know the origin of the phrase 'Not that old chestnut'?

Brewer's Dictionary of Phrase and Fable and the 'Facts on File' Encyclopaedia of Word and Phrase Origins agree on the following explanation:

In a play called 'The Broken Sword', by William Dimond, produced at Covent Garden in 1816, a character called Captain Xavier is always repeating unlikely stories about his exploits. On one occasion, talking to a character called Pablo, he mentions a cork-tree. Pablo corrects Xavier, saying that the tree was a chestnut, and 'I ought to know, for haven't I heard you tell this story twenty-seven times?'

The play was soon forgotten, but many years later in America, an actor named William Warren Jr recalled this episode at an actors' dinner, where another speaker had told a stale old joke. The actors who were present picked the phrase up, and 'an old chestnut' became a synonym for 'an old joke'.

The explanation involving a joke about a cork tree is itself an old chestnut. Consider: 1)the term "old chestnut" is commonly heard in the phrase, "trot out the old chestnut" 2) an old chestnut is a hackneyed expression 3) the use of "hackneyed" to mean over-used or worn out derives from the horse-drawn cabs or "hackneys"--often shabby affairs drawn by broken-down horses--of 19th century cities 4) a "chestnut" isn't just the thing that falls from trees; it's used to designate a certain color horse--"my money's on the chestnut." Hello! What pulls a hackney?--an old chestnut!