First of all, there's joint snapping, and then there's bone breaking.
Joint snapping involves hyper-extending the joint until the tendons holding the joint together snap, or the end of the bone pops out of its joint and becomes dislocated. It's painful, and it can damage the ends of bones at the cartilage. It can rip tendons and muscles. And it's relatively easy to do compared with actually breaking a bone.
Then there's bone breaking. That's where you snap the bone itself, either fracturing it or breaking it in two. This often is accompanied by joint dislocation and tendon / muscle damage.
Both joint snapping and bone breaking are incorporated into most martial arts. And it's similar to vital point striking, in the sense that it should already be in the martial art and requires no additional or supplementary skills to learn.
Let me give you an example. You can see standing arm-bars in most martial arts. You grab a hold of someone's wrist and extend his arm outwards so that his elbow is now locked. Then you keep that hold while pressing down on his arm at the elbow or mid-upper arm. That will hyper-extend his elbow and allows you to get leverage to push him down to the ground, perhaps with some extra foot-work.
But what if you didn't just apply leverage to that joint while taking him down to the ground? What if you did a hard, heavy hammer-strike to the point just above the elbow? Then that joint can easily snap. And you can still take him down to the ground with it.
So this is really just a difference in how much force you apply. There's very little, if any, change in the mechanics of the technique. You're changing a push into a strike. But mechanically, it's doing the same thing.
But that's not actually breaking any bones. It could if you do it hard enough, sure. But it's not optimally designed to break bones.
There are martial arts that sort of specialize in joint snapping and breaking bones. Or at the very least, they do train to do it very specifically, as opposed to martial arts that merely imply it and don't develop the kinds of strikes needed to accomplish bone breaking. And they even will often use it as a marketing tool to attract students. Haha.
Two martial arts you might want to look into for this are: Okinawan karate (naihanchi kata bunkai), and Koppo Jutsu (which is often linked to Bujinkan budo taijutsu or Koto Ryu). Of course, these are just off the top of my head. Pretty much any martial art will have these kinds of applications built into them, but perhaps there are some that make it their point to train to be able to do this and explicitly teach it rather than merely implying it.
Okinawan karate comes from White Crane and Incense Shop Boxing kung-fu and other styles. Bujinkan takes from classical Japanese samurai ryu, and most of them do have their favorite bone breaking techniques in them. So you don't necessarily need to look at karate or Bujinkan. You can start to notice these techniques everywhere. But I gave you those two styles as a starting point.
One of the things you'll notice about styles that do train people in how to break bones is that they'll emphasize conditioning their strikes. So they'll do a lot with punching bags and punching hard objects (I've even seen people slamming hands and fists against brick walls). The goal is to be able to hit hard and heavy without breaking your own hand, wrist, foot, or arm. If you're attacking your opponent's shin bone by performing an oblique stomp downwards and on an angle, then you have to be able to do that without injuring yourself. It's not easy, because their bone is as hard as yours.
And just like with vital point strikes (strikes to weak areas on the body), the technique you use has to be able to work even if it doesn't break bones or snap joints. So for example, that standing arm-bar that I wrote about above will work even if his elbow doesn't snap, even though you were trying to snap that elbow. It's okay. It doesn't have to break the bone or snap the joint in order for it to be an effective technique.
If, on the other hand, you're just focusing on breaking bones, and you don't care that the technique still works even if it doesn't break bones, then you're probably training the wrong way. And most martial arts know this and don't specifically target breaking bones. It's gotta work even if the bone doesn't break.
Hope that helps.