Whatever JRR Tolkien might have said about the veracity of his invented history of Europe, it’s quite obvious that the Middle Earth of the Lord of the Rings is quite distinct from our own world - the War of the Ring did not, in fact, take place in a lost epoch of European history, and Bullroarer Took did not actually win the Battle of Greenfields and invent the game of golf at the same time. The world of Arda and the events that take place within its bounds are plainly an “imaginary world” (Letter 131). Still, I believe that Tolkien would argue that this imaginary history is, in many ways, no less “real” than our own.
Or, as one might also say, our history isn’t quite as “real” as we sometimes imagine it to be. As he discusses in his Notion Club Papers, simply by virtue of the frailty of human memory, our conceptions of past events can never be wholly faithful to what truly transpired. “What do you know about “true past events?” asks Tolkien through the mouth of Ramer. “They are all stories or tales now, aren’t they, if you try to bring them back into the present?” Past events necessarily becomes “stories or tales” as the facts behind them are distorted by time, memory and continual retelling, until “history” becomes only “story”; its cease to be real, and living individuals become mythological characters (it’s no longer “his” story, get it? :). History is a narrative constructed out of disparate myths and clues passed down to us - and who can say that this invented story is any more real than Tolkien’s? As Tolkien says again (now through Jeremy), “What is invented is different from mere fiction; it has more roots.”
And part of what makes the Lord of the Rings so compelling is that Tolkien uses the same “ingredients” that form our mythological conceptions of history to create his Arda, drawing parallels, grounding it in reality and making it even more difficult to separate his world from ours. Many Middle Earthian historical elements seem to mirror folk memories of real-world events; the “sea kings” are an apt analogy for the coming of the Romans to England; the Drúedain reflect the Medieval “woses”, who, to Tolkien’s mind, are themselves mythological incarnations of post-Conquest, remnant native Britons. I believe that what Tolkien was trying to do here - by planting the roots of his invented world in our own history - was to create a sort of alternate, mythic history, one that could easily run alongside Arthur and Sigmund.
That being the case, there remains the question of how much value one should place in this sort of legendary past over the real, concrete past of the learned historian (of which Tolkien was one). Again, Tolkien voices his opinions on the matter in the Notion Club Papers: “Of course, the pictures presented by the legends may be partly symbolical, they may be arranged in designs that compress, expand, foreshorten, combine, and are not at all realistic or photographic, yet they may tell you something true about the Past.” On a practical level, of course, folk tales can be of great value to philologists as repositories of words and place names, as Tolkien mockingly illustrates with Farmer Giles. But they also often contain within them seeds of truth, seeds which can help you get to the men and actions behind the legends. "There are real details, what are called facts, accidents of land-shape and sea shape, of individual men and their actions, that are caught up: the grains on which the stories crystallize like snowflakes. There was a man called Arthur at the centre of the cycle."
T.H. White seems to have been of a similar mind to Tolkien on the value of myth vs history. In The Once and Future King, he calls “real” figures like King Richard the Lionhearted “imaginary”, and paints Arthur and his Court as having been real, living people. I see this as less of an assertion of the reality of Arthurian legend and more a commentary on the subjectivity of history, and the impossibility of fully delineating historical fact from myth. The lionized (no pun intended) Richard that we know today is a legendary figure, too, no more real than Lancelot or Guinevere. A king named Richard may have once ruled, but what we have now is a legendary man and a kingdom “glorified by the enchantment of distance in time.” (Letter 183).
For a last note on what Tolkien believes separates reality from fantasy, I would look to his commentary on right and wrong within the confines of his story and history (183). The central conflict in the Lord of the Rings is not, what he calls, “political”; Frodo’s cause in destroying the ring is far-reaching, a struggle not of man against man but of all humanity against tyranny. This is the stuff of “wonder-tales”, Tolkien says, not of reality. “Of course in 'real life' causes are not clear cut — if only because human tyrants are seldom utterly corrupted into pure manifestations of evil will.” Still, history often turns these very political, “real life” causes into the kind of mythic struggle that we see in LOTR; war stories valorize the victor as the Good, and villainize the enemy so that he becomes Pure Evil, and not the ordinary man that he is. Lord of the Rings, as a saga of war, is fantasy - but then so are many of our ancient war stories.
All of which is to say that Tolkien may not have seen a valuable distinction between supposedly “factual” history and “mythic” history - facts are distorted and become the seeds of myth, which we must parse these stories to get at the truth of it all. History becomes legend, legend becomes myth.