The term "tinhorn" in the Wild West was used to refer to someone contemptible who would claim or pretend to have greater influence, wealth and ability than they actually had. 

Tinhorn actually refers to a dice rolling device.  A hazard chute was a horn or cone shaped device used in the dice gambling games Grand Hazard and Chuck-A-Luck.  The chute would have dice dropped into one end and would often have inclined trips inside to ensure the dice rolled out of the other end randomly.  The original hazard chute was made of leather but small time operators, with limited resources, would use one made of tin.

Antique Tinhorn

It was the common opinion at the time that items made of tin were poor imitations of better quality goods.  "Tinhorn gambler" was a derogatory phrase for cheap Chuck-a-Luck operators and "tinhorn sports" - the players who would gamble small stakes on the game.

Chuck-A-Luck was a simple and easy game to set up and operate, only requiring a basic layout, three dice, and a hazard chute.  It was the small time operators game of choice on riverboats, in cheap saloons and in cheap gambling houses.  These operators often tried to make up for their low status by wearing fancy clothes and putting on a loud, showy demeanour, while in fact they where at the very bottom of the social scale along with the three-card-trick conmen and the keepers of the cheapest bars.. And so "tinhorn" was used to refer to anyone with crass pretensions above their station.