One can say without contradiction that King Lear could be the greatest of all Shakespeare's tragedies, but that it perhaps makes the least "conventional" sense from a purely dramatic standpoint. This is why modern commentators have sometimes likened it to the Theatre of the Absurd of Beckett, Ionesco, and others. Though it may have been less performed before the twentieth century, earlier critics such as William Hazlitt still recognized it as a masterpiece.
Much of the action of King Lear has been seen by some critics as far-fetched or improbable. To give just two examples: Cordelia, though she's the only daughter who loves Lear, is strangely reticent after the other daughters have (falsely) poured out their affection for him, and Lear overreacts against her. In a similarly improbable turn, Gloucester immediately believes Edmund's lies about Edgar and flies into a rage. What is it, however, that gives these developments a sort of higher truth than the photographic realism of, say, a movie or TV show?
The story of Lear is symbolic of the irrational nature of the world and of life in general. Readers and audiences probably recognize this fact without necessarily being objectively aware of it. The play seems to take place in a dream land, and our sense of this is heightened by its setting in an ancient, quasi-mythic Britain. The fact that Shakespeare uses modern place names like France and Burgundy and titles such as Duke paradoxically enhances the atmosphere of unreality. These elements may have all contributed to the difficulties of doing conventional stagings of the play, but in our time they are seen as factors contributing to its greatness.
For further reading, I would suggest George Orwell's essay "Lear, Tolstoy and the Fool," which analyzes the play in the light of these issues.