A good accounting of the sword's usage for Vikings (in the early Middle Ages) is found in Hilda Ellis Davidson's The Sword in Anglo-Saxon England". The short answer for the question in the text—whether "it was easy to behead someone or cut one of his limbs"—is: yes, swords could behead or dismember relatively "easily", and, it appears, often did. /u/DeusDeceptor goes over the executioner's effectiveness, which is a fairly good marker as well.

Sharpness is definitely a factor in the removal of limbs, and the Vikings, for instance, often gave their swords to specially trained men for sharpening. Additionally, however, the weight of the weapon, strength of the metal, and strength and skill of the blow all factor into potential damage of the blow. Bear in mind that a "real" sword is a sturdy, hefty weapon that ties the edge-holding ability of metal and substantial weight of the blade with the leverage available to a swung weapon. The sword's millennia-long dominance in melee combat was well-earned.

While the amount of force needed to cut through a human body, particularly when armor is involved, is not negligible, strong warriors could certainly do serious damage: bodies have been found in medieval archaeological sites with skulls chopped in two with a downward stroke. One even apparently had both legs removed by a single blow of the sword. In records of duels, in which Vikings traded heavy blows one after the other, wooden shields are often broken by a single sword strike; there are records of four or five shields being lost by each participant in this manner, all in a single duel. Throughout the sagas, warriors use the sword (with almost comic regularity) to remove heads and limbs, and occasionally are recorded slicing through torsos.

The executioner Frantz Schmidt worked from 1573 to 1617 in Bamburg and Nuremberg and recorded an extensive, if somewhat sparse account of his career in a diary. Schmidt recorded in his diary 187 executions with the sword. He records that only 4 of these executions required a second stroke. The diary does not expound on this much, it only denotes that they were "botched". This is a 2 percent failure rate, and, while it would have been horrible to be among those 4 people, there is a fair chance that many of the "successful" ones would have undergone some other corporal punishment as well, so they aren't getting of entirely without torment.

Balancing the public spectacle of fair corporal punishment with unfair cruelty was a difficult act, and botched executions could result in the crowd turning against the authorities. In some German towns executioners were permitted three failures before they could be seized by the crowd and exchanged with the accused.

As the purpose of public executions was to legitimize a core message of state authority and religious redemption/condemnation, competent executioners were an absolute necessity.