Could dragons that look exactly like some of the art we see today exist, with wings and four legs and so on...
No.
Could dragons as a concept have come about as misunderstood re-tellings of stories about real creatures and also fossil remains ... yes absolutely.
Humans have an inherent need to tell stories, to pass information in that form, which is why we still go to see plays (even plays we've seen before or read and know the whole story) and watch dramas on tv and read books and watch films..
Its why forums exist, we're all telling stories, some truer than others.
When those stories can only be passed on by telling and listening rather than writing down and then later, by poorly translating loose written language, those stories change, details are fudged..
Humans also like to embellish. Its got to be bigger and better and more ferocious and terrifying, we like to add detail that may not be strictly necessary, or accurate.
Historically, when we've depicted anything with images, we've embellished it, from Lord Thingies cows and horses that were far taller or fatter or more magnificent than their real life counter parts, to Joe the Sailors sea-monster, which swelled to 100s of feet longer and bigger and more vicious than the stinking remains of the collossal squid he saw washed up on a beach...
It isn't hard to see how stories grow and details are mixed with misunderstandings.
One person describes a huge reptile, explains that you don't want to get the animals saliva on you as it burns when it gets into an open wound (which you naturally have if you wrestled it to the ground)... becomes another persons 'it breathes fire'..
And like every angler who ever caught a tiddler that became a whale by the time he got home... all these things were significantly larger in the retelling of the tale.