The Full Monty, meaning ‘complete’, or ‘the whole thing’ (more recently with nudist connotations) could have originated from a number of competing claims. It is typically attributed to the tailoring business of Sir Montague Burton. In this context, a complete three-piece suit, i.e. one with a waistcoat, would be ‘the Full Monty’.
Sir Montague was certainly famous enough to have sired the phrase, but personally I prefer to believe that it owes its existence to Field Marshall Montgomery, leader of the infamous Desert Rats. It might have been something to do with his habit of wearing his full set of medals, or his insistence on his troops eating a full English breakfast every day, or his rigorous training regime. Take your pick.
Incidentally, the American equivalent – ‘the whole nine yards’ – may also have originated from the Second World War, where aircraft gunners would apparently give their enemy ‘the whole nine yards’ by firing their entire ammunition belt (as opposed to the normal practice of short, controlled bursts) the belt being nine yards long. However the evidence in this case is more than a little shaky.