Depends on the airplane.
Military fighter airplanes have both G limits and total load limits. At high gross weights, the G limit is reduced from the "Max G."
Most civil airplanes have only a G limit published.
The long-term issue is one called "low cycle fatigue." In fighters, the number of times an airplane is stressed to X or Y Gs is significant. The greater the number of Gs, the fewer number of cycles can be safely attained over the lifetime of the airframe, and that number goes exponentially lower for each increment of higher G.
In carrier-based aircraft, arrested landings are a factor. In civil airliners, Takeoff and Landing "Cycles" are a factor.
Exceeding a set number of cycles of G, landings, or other limiting stress may not guarantee that something will break, but will increase the probability beyond the design safety factors -- and also increase the probability something will break at lower stress.