Possibly, but undetermined.
Arthurian legend comes for a variety of contradictory (sometimes self-contradictory) sources so it's near impossible to make solid claims about the legends.
One common claim is, despite what people think and what the movies tell us, that the Sword in the Stone isn't Excalibur. The theory states that Excalibur is the sword given to Arthur by the Lady in the Lake and that the sword Arthur drew from the stone is unnamed.
As I previously noted there are too many sources to make any general claims about the legends, so let's look at a specific legend: Thomas Malory's Le Morte d'Arthur. Malory took many of the existing legends and combined them into a single cohesive story which has become the got-to text for Arthurian legends ever since. Except it wasn't particularly cohesive which is why we end up in Malory with not one but two swords called "Excalibur."
The sword in the stone is unnamed, and although it's referred to being in a stone a number of times in the test, when it is described in detail it is a sword in an anvil attached to a stone. It is unnamed and Arthur pulls it from the stone, something no one else can do and so he's made King of Britain. Except for a bunch of mainly northern kings who don't like the idea of a beardless boy of low blood being king. So they turn up and the post coronation feast with their armies and lay siege to Arthur and his men for fifteen days, until finally Arthur leads his knights into battle. Near the end of the battle Arthur is dehorse, but four of his knight get him remounted:
("Then he drew his sword Excalibur, but it was so bright in his enemies' eyes, that it gave light like thirty torches. And therewith he put them a-back, and slew much people.")
So at this point, right after his coronation, Arthur has a magic sword called "Excalibur." It isn't explained where it comes from. No other swords are mentioned, so it could be the Sword in the Stone.
However, a few battles later, after the rebel kings are defeated and Arthur's mother Igraine finally turns up to confirm that he is in fact rightful king, Arthur comes across King Pellanore and after an arguement over a horse is soundly beaten, especially when the sword he is using breaks in two. This sword is unnamed and seems extremely unmagical. Merlin then takes Arthur off to meet the Lady of the Lake where she gives him a new sword.