A safety valve system for use in a subterranean well can include a safety valve protector connected downstream of a safety valve, whereby when closed the safety valve protector reduces a flow rate through the safety valve and prevents displacement of an object through the safety valve protector to the safety valve. Another safety valve system can include a safety valve protector which, when closed, reduces a flow rate through a safety valve, and the safety valve protector closes in response to the flow rate through the safety valve being above a predetermined level. In another safety valve system, each of the safety valve protector and the safety valve comprises an actuator, the actuators being connected by a line, and a signal transmitted by the line causes the safety valve protector actuator to close the safety valve protector, and then causes the safety valve actuator to close the safety valve.