The OED traces cabal to French cabale and ultimately to qabbalah, with the sense of secret plotting current by the 1640s. The Clifford, Arlington, Buckingham, Ashley, Lauderdale ministry genuinely held power from about 1667, and contemporary satire exploited the acronym fit, which the OED notes fixed the modern sinister flavor.
The popular version claims the word was invented from the ministers' initials, a backronym. The dates kill it: cabal in the intrigue sense appears in print before the ministry formed. The ministers did not coin the word; they tattooed it.
The Qabalah connection means the word for conspiracy descends from the word for esoteric wisdom: the secret teaching and the secret plot are the same noun seen by different crowds. Lexicographers note the irony without commenting further.
A mystical word and a coincidental acronym collided and neither survived unchanged. How much of meaning is pileups like this?