Breaking a Sword
Edge damage is a big thing among swordfighters. It's a given that any kind of sparring is going to result in your weapon taking a beating. But it's an interesting question to ask - is it possible to break a sword, like they do in the movies? Is it possible for the One Special Sword of Power to slice straight through another sword? The correct answer is 'LOL NOPE' assuming the sword in question has been made competently and we're talking about actual weapons governed by the laws of physics, and not crazy magical nonsense.
Let's talk about Steel
So here's the thing: good steel is not a modern invention. The efficient mass production of steel is a new thing, yes, but actual steel - the alloy of iron and other elements that increase its hardness or strength - is at least four thousand years old. I'm saying this specifically because I want to impress on you the fact that medieval weaponsmiths were not messing about, half-assing their work, or making stuff up; they had the benefit of a long tradition of steel production and sword-making to drawn on. They were good at what they did and they understood the material they were working with.
The bottom line is that it takes quite a bit of force to just chop through steel, especially steel that's been forged and hardened and is designed from the start to handle such force. I've blathered on before about the balance between hardness and flexibility in swords, but the general idea is this:
A sword must be hard enough to deliver force effectively, and flexible enough to dissipate force so that it doesn't shatter on impact.
Breaking a Sword
In medieval times, actually breaking a sword into pieces - as in, your sword is now a collection of sharp bits - was not an Actual Thing that could happen in combat. What was more likely was that swords would become notched and blunted and chipped. Those made out of particularly bad steel might not hold an edge at all, and be far less effective. It would also have been possible for a badly made sword to fracture along a pre-existing weakness in the blade.
There is a particular trope of either the Sword of Power(TM) smashing through its opponents, or the Sword of Power(TM) itself being shattered and having to be reforged, Narsil-style.
This is most definitely bullshit. Swords do not break like this. Glass does, or maybe breadsticks, but neither are suitable for combat.
Cutting Through a Sword
In the modern world, you'd cut through steel using a saw, or a grinder, or a cutting torch. (Steel is hella tough.) In medieval times, smiths used shears or a hammer and chisel to cut metal. All these things involve time and effort, meaning they're not going through steel in a single blow.
Probably the best reasoning behind the idea that swords were not cut through like butter is the fact that medieval armor was basically impenetrable to cutting by a sword. The same steel was available for manufacturing both armor and weapons, and plate armor was a very effective defense against the longsword. But even outside of that, we can see that the physics of swordplay does not lend itself to a sword being cut by another sword.
This all comes back to how energy is dissipated when two swords make contact. Ideally, a sword dissipates its energy in delivering a blow to a target; when parrying another sword, energy may dissipate in many ways depending on how both swords are moving through space.
They can bounce off each other.
They can slide against each other.
One can yield against the other.
In order for a sword to cut through another sword, they must connect and enough energy must be transmitted into one blade to make it happen. You can imagine that, if a person were actually strong enough to generate enough force in the swing, all the other methods of energy dissipation must not happen, or else there will not be enough left to complete the cut.
So their opponent must take the blow on the edge of their sword, the swords must not slide or bounce, the cutting sword must be thin enough to pass through the other but not so thin that it can't transmit force effectively on that plane, their opponent must not move their sword, the cutting sword must be much harder than the other but not so brittle that it shatters...
After a point, the physics becomes untenable. I won't say it's impossible - stranger things have happened - but I think it might as well be.
So here's the thing I've noticed about these two tropes, one where a sword is shattered and one where a sword is cut through: they require completely opposing real world properties. For a sword to shatter, it must be very hard and brittle, like glass. For a sword to be cut in half, it must be soft.
I find it interesting that the tropes of sword-breaking implicitly understand that a sword being too hard or too soft is equally terrible. I like to imagine that this is a long cultural legacy from the time when swords were used for their real purpose. It's nice to think that the narrative nonsense that pops up frequently in our stories does actually have a basis in reality.