A bottleneck essentially means one component will be maxed out (in it's ability) before the other component can get close.
Can result in stuttering, FPS drops etc - worst cases, some lockups or slight freezing, but should not cause crashes.
In this case, your FX4300 would reach 100% usage and have no more to give before a 960, RX4xx or GTX1060 gets close to maxing out.
A bottleneck, in & of itself, will not cause system crashes.