How close do the natural frequencies need for this to happen? There must be some relation between how tightly coupled they are and how close their frequencies must be. Is it a sharp cut-off? What happens near the breaking point?

Easier to explore in electronic systems than kinetic ones, as we did in a lab in college. You get some fun phenomena, with chaos and fractals arising: https://en.wikipedia.org/wiki/Arnold_tongue

Good points. This would certainly be interesting to figure out.