Invented at the card table
John Montagu (not Montague) was Earl of Sandwich (Kent, England) och the dish definitely got it's name from him - according to Morton it wasn't called anything in particular before that. But he didn't invent it.
... People had been wrapping slices of bread around bits of food for thousands of years before the earl came along.
Another piece of supposed fact in this case is considerably more subtle: that the sandwich was invented at the gaming table, allowing the earl to keep playing without interrupting for a cooked meal.
The oldest found source for the word sandwich is from 1762. The wide-spread tale about it's coinage comes from a diary written by a Frenchman who visited London in 1765:
A minister of state passed four and twenty hours at a public gaming-table, so absorpt in play that, during the whole time, he had no subsistence but a bit of beef, between two slices of toasted bread, which he eat without ever quitting the game. This new dish grew highly in vogue, during my residence in London: it was called by the name of the minister who invented it.
P. J. Grosley
Which one historian has commented upon as follows (never mind the authorship of the dish, which disagrees with Morton's quote above):
Grosley's book is a piece of travel literature. There is no supporting evidence for this piece of gossip, and it does not seem very likely that it has any foundation, especially as it refers to 1765, when Sandwich was a Cabinet minister and very busy. There is no doubt, however, that he was the real author of the sandwich, in its original form using salt beef, of which he was very fond. The alternative explanation is that he invented it to sustain himself at his desk, which seems plausible since we have ample evidence of the long hours he worked from an early start, in an age when dinner was the only substantial meal of the day, and the fashionable hour to dine was four o'clock.
N. A. M. Rodger
This would, then, be a rumour, created to put Sandwich in a bad manner; if that is so, it would probably not have been the only one (true or not), since he was indeed involved in several scandals of various kinds. In the 1770's, by the way, he became First Lord of the Admiralty, and was responsible for the Royal Navy during the American War of Independence. But this was well after the sandwich.
Slightly related: Cook thought of John when he coined the name Sandwich Islands. It would eventually be replaced with the native Hawaii.