Depending on the quality of the metallurgy at work, breaking a knife can be an issue. I don’t know if that’s the case with the Fairbairn-Sykes specifically, though.
I was once advised, “if you use you’re knife as a screwdriver, you’re going to break it.” I can’t remember if that was directed at me at the time, but I’m definitely guilty of doing this. Thing is, knives aren’t designed to be torqued or flexed. The blade will stab, slice, and cut, but twisting and bending are things it’s not really designed for.
Now, “twisting the knife” is a real thing, and in soft tissue, it’s nasty. But, if you wedged it in a bone, and twisted, it’s entirely possible you could shear the blade apart. And, having recently dismembered a screw in the wall, my faith in modern metallurgy is a little shaken.
Is your character strong enough to do it? Yeah, they should be. If the blade gets caught on metal, or in a bone, and your character twists it or flexes it, there’s a real risk of breakage. That is a risk, not a certainty. I’ve got a knife around here, somewhere, with a thirty degree twist on the tip, (from using it as a screwdriver) and at least one cooking knife that lost about a millimeter it’s tip in a bone.
Now, the Fairbairn-Sykes was designed to survive the rigors of combat, so I don’t know exactly how abuse resistant that knife was. Or other high quality combat knives, in general.
Also, this probably won’t apply, in the same way, with the polymer and carbon fiber knives that are kicking around these days.
Now, what won’t happen is the tip breaking off in soft tissue. Any decent knife training should have taught your character to avoid burying a knife in a rib. But, if you abuse a knife enough, it will break.