Getting stabbed comes to mind.
I’m tempted to leave it at that, but I should probably go into some very basic anatomy.
Your soft tissue (muscles, tendons, and ligaments) form what you could describe as a kind of pulley system across your skeletal structure. They contract and expand, based on electrical impulses from your nervous system, and this is what allows your joints to move, and tie your skeleton together.
When the soft tissue tears or is cut, then the system starts to fall apart. The classic example of this is the Achilles tendon, which, if destroyed will make it impossible to walk on that leg, you simply cannot control the ankle anymore, but even just cuts into the tissue will impair you.
So, getting stabbed or slashed will cause you to lose some function in that limb. How much will be determined by exactly where and how deeply you were hit. But, it doesn’t really take that much damage to the wrong places to completely immobilize someone.
This is a large part of why “first blood,” is such an important concept with sword fights. It’s not just that one of the combatants is losing blood, and at a disadvantage from that. Their injury will also (likely) impair their ability to fight. Inflicting the first injury in a duel is very significant.
As for bleeding out, all that requires is that you lose blood faster than your body can clot the injury. Usually we think of this in the terms of arterial bleeds, where you’ll be dead in minutes at most, but, it’s important to remember that blood loss is cumulative. So someone can bleed to death from non-arterial injuries if they can’t keep the bleeding under control, and their body can’t clot the wound.
This is why sword and knife fights don’t always focus on getting a single killing strike. It’s entirely possible to wear an opponent out through bleed induced attrition.
We’ve discussed hypovolemic shock before. It’s worth remembering, once you’ve started losing serious amounts of blood, you’re not going to be particularly coherent. So, if you were planning for a character to have some deathbed confession while bleeding to death, that’s not really the way the human brain works when it’s losing that much blood.
Incidentally, this also applies to the fight itself. As a combatant loses blood, they’ll pass through those same stages. (Anxiety, transitioning to confusion and disorientation at ~30% blood loss.) These have immediate implications in a fight, where their opponent is looking for ways to exploit any weakness or vulnerability. Also, because the fight is constant physical exertion, they will lose blood faster than they would if they were resting.
Also worth remembering, it’s entirely possible to “win” a duel and still bleed to death afterwards.