One issue that gives paleontologists (and scientists in general) fits is the impossibility of proving a negative. For example, no one can demonstrate, with 100 percent certainty, that every single Tyrannosaurus Rex vanished off the face of the earth 65 million years ago; there's an exceedingly slim chance that some specimens managed to survive, and are happily breeding even now on a remote, and still undiscovered, version of Skull Island.
This isn't simply a rhetorical issue. In 1938, a living Coelacanth--a prehistoric fish believed to have gone extinct at the end of the Cretaceous period--was discovered off the coast of Africa. To evolutionary scientists, this was as shocking as if a live Diplodocus had been discovered in Siberia, and caused some quick rethinking about the casual use of the word "extinct." (The Coelacanth isn’t technically a dinosaur, of course, but the same general principle applies.)
"Living Dinosaurs" and Cryptozoology
Unfortunately, the Coelacanth sensation has bolstered the confidence of modern-day "cryptozoologists," investigators (not all of them scientists) who believe that the so-called Loch Ness Monster may actually be a supposedly long-extinct plesiosaur, among other odd theories. Creationists, too, are especially eager to prove the existence of living dinosaurs, since they believe this will somehow invalidate evolutionary theory (which it won't, even if that mythical Diplodocus is ever found).
The fact is, every time reputable scientists have investigated rumors or sightings of living dinosaurs, they’ve come up completely dry. Once again, this doesn’t establish anything--that old "proving a negative" bugaboo--but it is persuasive evidence in favor of the total-extinction theory. (A good example of this phenomenon is Mokele-mbembe, a putative African sauropod that has yet to be conclusively glimpsed, much less identified, and probably only exists in myth.)
Many people also cling to the idea that the "dragons" mentioned in the bible (and in European and Asian folk tales) were actually dinosaurs--and that the only way the dragon myth could have arisen in the first place is if a human being witnessed a living, breathing dinosaur and passed the story of his encounter down through countless generations. This is complete nonsense, of course--for more, see this investigation of dinosaurs and dragons.
Are Birds Living Dinosaurs?
In any case, a question like "Did the dinosaurs really go extinct?" may be missing the point. Any group of creatures as numerous and diverse as dinosaurs were bound to pass off a huge chunk of their genetic material to their descendants, no matter what form those descendants took. Today, paleontologists make a convincing case that dinosaurs never really went extinct at all; they merely evolved into birds, which are sometimes referred to as "living dinosaurs."
The "living dinosaurs" motif makes even more sense if you consider not modern birds--which are mostly a tiny, docile lot compared to their distant ancestors--but the gigantic "terror birds" that lived in South America millions of years ago. The biggest terror bird of them all, Phorusrhacos, measured about eight feet tall and weighed in the neighborhood of 300 pounds--and it also hunted much like a middleweight theropod dinosaur of the Jurassic or Cretaceous periods!