What is certainly known about the possibility of a real King Arthur . . . is that it's possible he existed. That much is noncontroversial.
However, I'm of the "where there's smoke, there's fire" school of thought, meaning I believe the "possible" can be pushed to become "probable." Here's the skeleton of my argument.
The Battle of Badon is generally viewed as a real historical event.
Not only is it mentioned in a nearly contemporary source (Gildas),
but there's also confirming historical and archeological evidence
demonstrating a temporary halting of the Saxons' westward incursion,
for a couple of generations. This interruption coincides with the
documentary timing of Badon.
Someone was no doubt the overall commander of the Brythonic forces.
This person would be our theoretical "Arthur."
The only name ever explicitly given, by any writer, to this
commander is "Arthur." No competing name has ever been put forward
by any ancient source - unless we take Gildas's mention of Ambrosius
Aurelianus, several sentences before the mention of Badon, as
indicating the name of the commander. However, in the immediate
context Ambrosius can be understood as instigating the Brythonic
resistance to the Saxons one or two generations before Badon,
rather than necessarily having been the one who personally led the
troops at Badon a generation or two later.
On balance, these considerations incline me to believe Arthur really existed - and by that very name; it wasn't merely a title or nickname. However, since in the immediate several generations following Badon there were several "Arthurs," all of them Irish, and since there were multiple Irish settlements on the island both before and after Badon, I believe (for now!) that Arthur was of Irish stock (though perhaps also partly Brythonic). This would be consistent not only with the "Irishness" of his name, but also with the Historia Brittonum's distinction between Arthur and "all the kings and military force of Britain . . . . And though there were many more noble than himself, yet he was twelve times chosen their commander": i.e., Arthur was not himself a Brythonic king, nor did he match their tribal/political nobility.
In this connection, an intriguing little piece of relevant history was uncovered at the Roman and subRoman Viroconium (adjacent to the village of Wroxeter), sometimes argued to be "Camelot" due to its early-6th-century rebuilding under a Brythonic (or Irish???) warlord. At that time it was the island's 4th-largest city. "A tombstone found [at Viroconium] in 1967 in ploughing bears an inscription to Cunorix. The use of the word macvs for “son of” is an Irish form and dates the stone to the late 5th c. or later. Probably the man was an Irish mercenary employed by the citizens to protect them from wandering bands of brigands." (Richard Stillwell, et al, eds., "VIROCONIUM CORNOVIORUM (Wroxeter) Shropshire, England," The Princeton Encyclopedia of Classical Sites, http://bit.ly/290wEKY [emph. mine])
The single-best argument in favour of Arthur's existence that I've yet seen is Christopher Gidlow's The Reign of Arthur (The History Press, 2007). I can't recommend it highly enough.