Bumptious is used in the sense of conceited. There is a connection to the word bump in the expression "puffed up with conceit" where puffed up refers to bump.
Puffed up (and puffy) -as an adjective- has a similar sense to bumptious as well.
This connection is also mentioned in one of the earliest dictionaries I could find. (Glossary of Northamptonshire Words and Phrases, Volume 1
By Anne Elizabeth Baker - 1854). Another form of the word is bumshus.
An earlier dictionary mentions it as a word from Norfolk and Suffolk dialects. (A General Dictionary of Provincialisms By William Holloway - 1839):
Another dictionary suggests that it might have emerged from or gained usage through college jargon (as college humor) in Cambridge. (A Collection of College Words and Customs By Benjamin Homer Hall - 1856):
Here is the earliest description of bumptious from a Cantab in the context of chess. (from The New Monthly Magazine, Volume 3; Volume 39 - 1833):
In sitting down to play, take notice how far your adversary troubles himself about arranging the board and men, or whether he obtrudes all the preliminary settlement upon yourself. If the latter, and if he makes you set a good part of his own men for him, you may be sure he reckons himself something too good for you, and stands high in his own esteem. At Cambridge we called such a man bumptious. It attends him in all his actions through life.—" L'âme n'a pas de secret que la conduite ne révèle. L'amour-propre est le plus grand de tous les flatteurs. "