Namely with swords. Could something like a one handed arming sword completely sever something like an arm? Or could a longsword? Assuming the target is unarmored, the weapon is considered quite sharp for what it is and what is available to sharpen it. And namely, but not restricted to Europe during the medieval time (5th to 15th century)
This is a good question and we actually have quite a bit of data on injuries, a lot of which comes from the 14th century battlefield at Visby. However, we know of others: there's also a Late Roman era burial at Zadar and a 7th century burial in Britain (the name escapes me but I think it's Chesterfield?) which all show similar patterns to the Visby battlefield.
Severing an arm is a matter of the strength of the hand powering the blade, and yes, it is entirely possible to sever an arm in a single hack with a handheld, human-powered weapon. The problem is that doing so isn't ideal. When we look at the patterns of injuries in battlefield graveyards, forensic archaeologists/anthropologists typically find that the majority of wounds were small, light blows to the arms and legs, with a later finishing injury to the head. Wounds to the thighs and chest are exceedingly rare at least in terms of the information we can garner from tool marks in bones.
So why weren't limbs regularly lopped off in combat? The first reason really only applies to after the arming sword developed out of the Roman ("Byzantine") Spathion in the 10th century and spread across Western Europe. Shortly after the invention of stereotypical "long swords" as we think of them, the development of a proper method of combat with such blades developed. The earliest written evidence for the techniques used in what we now call "Historical European Martial Arts" come from the 13th century AD, from Western Europe, where the earliest fragments of written instruction for sword combat can be found. Although we have evidence for proper swordsmanship training through offhand references in texts and art earlier, probably pointing towards an origin in the hoplomakhia of the professional Roman army. But I digress, this is my point; these techniques of swordsmanship result in quick, light confrontations where the wounds inflicted are typically shallow and to the limbs, since that's the easiest part of your opponent to hit, and you're trying to be as proactive as possible while exposing yourself as little as possible.
This gets into the second reason which is applicable for the entirety of the medieval period (638 CE-1453 CE) and before: fatigue. There's a lot of debate still as to exactly how battle operated in the ancient and medieval world, with many scholars suggesting that it was a continuous series of short engagements of the two lines who would fight and withdraw to recover, before moving out to fight again. The duration of this combat is thought to have been between about 2 and 5 minutes in each "skirmish", which is about the maximum amount of time a person can sustain continuous hand to hand fighting in those conditions. Yes some people can go for longer, but professional boxers or the like are not analogous to medieval combat. You're typically holding a sizable 10 to 13 pound shield which covers most of your body, and an 8 foot infantry spear. Your comrades are pushing behind you, pushing you forward, to keep the mass of soldiers coherent, and you're constantly moving your shield and spear proactively and reactively, both trying to keep yourself defended and find a brief opening to hit one of your opponents. In that situation, the overwhelming majority of your body is protected by your shield with only your head, lower legs, and outer arms continuously at risk unless you do something stupid and expose yourself. Your movement had to be quick, light, and fluid, so large, heavy blows like in hollywood films were essentially impossible. This is reinforced by the fact that in such a formation, you have little space for wide movements. A wound to the limb was just as effective as a wound to the core in medieval times, so any jab you could get in was worth the effort, and was usually effective, although most deaths occurred in the rout. After the battle you'd just go and finish off the wounded with a hand axe or other weapon. So this is largely why completely lopping off limbs was uncommon: it required too much space and energy for the situation you were in.
I can't speak for fighting styles in medieval Japan or East Asia, but I hope this answers your question for Western and Eastern Europe and the Mediterranean.