I've mentioned it before - the worst thing that can happen to a rider is a broken girth. If the saddle comes off - so do you.
Therefore, in some extreme riding sports, riders use an overgirth. This is an extra strap that goes over the saddle rather than being attached to it - you most often see this on eventers and racehorses. The theory is that you would have to be extremely unlucky to have both billet straps and/or the girth and the overgirth all break at the same time.
This polo pony, who looks half asleep while waiting for his chukka, is wearing an overgirth - the black strap seen running over the saddle. (Source: RRukat via Wikimedia Commons).