Although prehistoric animal bones are often put forward as an explanation for dragon beliefs, there is no proof of this. It is pure speculation. Indeed, the idea that dinosaur bones inspired traditions involving dragons has become a matter of modern folk belief.

While many things including dinosaur bones can put wind in the sail of tradition, there is no evidence that they could actually inspire any given tradition. Folklore doesn't tend to work that way: "Hey, this is something odd I just found, I bet it exists because long ago there must have been ... fill in the blank with a new creature [giants, cyclopes, dragons]." Such a scenario is only speculation, cannot be confirmed, and runs counter to what has been documented by folklorists and ethnologists. Instead, the process can work like this: "I believe in [fill in the blank] and look what I just found; I wonder if this is a remnant of those creatures." This option is not impossible, and it is easy to imagine. Unfortunately, enthusiasts often take the second scenario and transform it into the first with no evidence except their own imagination. For example, many people have claimed that mammoth skulls are the explanation for the cyclops motif, but there is absolutely no evidence to conclude that this was the origin of the tradition.

Dragons were generally believed to have existed, but the folk typically relegated them to distant times or places. There are no legends of people coming upon them in the wild. Even the celebrated observation in the Anglo-Saxon Chronicle of dragons being seen in the clouds is a literary report that was likely something the chronicler heard - a legend from a distant place. Whether we can regard that as an exception or not, folklorists working with pre-modern informants did not collect western/northern European legends about encounters with dragons. People did believe they once existed, and these entities do figure in folktales, because these fictional stories were often set in remote places and times - "Once upon a time, ...".

The folklore side of your question has been ably addressed, but there's also some trouble on the paleontology side. It's not impossible that people did find fossil remains before paleontology was an actual field of study, but there are certain practical difficulties, as u/MrPaleontologist explains. If anyone else would like to chime in on this topic, please don't let this post stop you! We always welcome new insights on any matter.