8 mm and below is small bore. The German 8x57IS was the biggest of the cartridges allowed in "small bore" competitions. I guess since the competition rules have moved on the definition may have changed.
8 to 10 mm is medium bore. You will find a lot of reference to the 375 H&H as a "medium bore" round.
Over 10 mm is big bore.
In America small bore is now synonymous with rimfire, although I don't know how that developed. In some countries "fullbore" refers to a service rifle competition (in 308 Win).