Yes. There’s an artery in the wrist and when it gets nicked, then the person who was stabbed will bleed out and they will do so fairly quickly. You can do the same with the upper arm as well. It’s also worth noting that when nerves, tendons, and muscles get cut (slashing the hand, slashing the bicep, slashing the triceps), they have more difficulty working and could, potentially, stop working all together.
This is part of the reason why knives are bad news bears and why even if you have a knife, you want to run away from someone else wielding a knife. Knives are precision work, the shortness of the blade means that the fight happens in much tighter quarters, making the strikes difficult to block, and is over much more quickly.
It’s also worth noting that when one does get stabbed or fights with an open wound, they are increasing the likelihood of death. Your heart rate elevates during physical activity, the blood keeps pumping through the body, and spills right out of the hole. The closer you get to an artery, the faster that happens. This is why pulling an object out of your body in the middle of the fight is a bad idea. If someone buries a knife in your back, you want to leave it there until the fight is over. It’s not the best solution and may continue to do more harm, but it will also keep the blood you need to live from spilling out until you can find a better way to contain the wound.