People often claim that this word originated as an abbreviated form of the Latin phrase sine nobilitate, meaning ‘without nobility’ (i.e. ‘of a humble social background’). Various accounts of the circumstances in which this abbreviation was supposedly used have been put forward: on lists of names of Oxford or Cambridge students; on lists of ships’ passengers (to make sure that only the best people dined at the captain’s table); on lists of guests to indicate that no title was required when they were announced.
The theory is ingenious but highly unlikely. The word snob is first recorded in the late 18th century as a term for a shoemaker or his apprentice. At about this time it was indeed adopted by Cambridge students, but they didn’t use it to refer to students who lacked a title or were of humble origins; they used it generally of anyone who was not a student.
By the early 19th century snob was being used to mean a person with no ‘breeding’, both the honest labourers who knew their place, and the vulgar social climbers who copied the manners of the upper classes. In time the word came to describe someone with an exaggerated respect for high social position or wealth who looks down on those regarded as socially inferior.
It’s quite possible that the phrase sine nobilitate may have appeared in one context or another, but it is difficult to see why it would have given rise to a word for a shoemaker.