In NRA competition, small bore refers to .22 rimfire rifles.
Years ago, a neighbor who hunted big game in Africa said the British system called small bore anything below 30 caliber. Medium bore was 30 to 40 and big bore was any caliber over 40.
There's probably a few different naming conventions. Different strokes for different folks. . . . . .
http://www.chuckhawks.com/medium_bore.htm
http://www.chuckhawks.com/intro_big_bore.htm