Here's a suggestion as to the origin of the expression: <<This expression comes from William Dimond's play, The Broken Sword (1816), in which one character keeps repeating the same stories, one of them about a cork tree, and is interrupted each time by another character who says "Chestnut, you mean . . . I have heard you tell the joke twenty-seven times and I am sure it was a chestnut.">>
From the OED:
A story that has been told before, a ‘venerable’ joke. Hence, in extended use, anything trite, stale, or too often repeated. Also attrib.
The little addition at the end of the definition (also attrib) indicates that the form chestnut excuse is acceptable.
The OED says that the origin is not certain, but makes reference to the same source as Bevj, and gives a later reference as well: 1888 in J. Hatton Remin. Toole, ‘When suddenly from the thick boughs of a cork-tree’ ‘A chestnut, Captain; a chestnut.’ ‘Bah! booby, I say a cork-tree!’ ‘A chestnut,’ reiterates Pablo: ‘I should know as well as you, having heard you tell the tale these twenty-seven times’.
As panjandrum says, a chestnut is something that has been said or heard too often, and so has lost its meaning, or its believability. In my experience, this is usually referred to as an old chestnut; the words go together to give it this meaning.
An old chestnut excuse then, would be an excuse that is an old chestnut, an excuse that is no longer believable because it has been overused and said too often in exactly the same way.
It is an unusual use ~ again, in my experience. "Old chestnut" generally used by itself, not as a modifier of something else.