No. Dinosaurs went extinct, approximately, 65 million years ago. Homo sapiens (us) emerged roughly 220,000 years ago. So, there is a huge gap in between the end of the Cretaceous Period (when Dinosaurs disappeared) and the emergence of our species. It isn't even close, really.
However, while most theories concerning Dinosaurs disappearing from the Earth tend to favor some type of mass extinction (I believe the asteroid theory remains the most popular), there is evidence that at least some dinosaurs evolved into modern day birds. To what extent and how many species may have evolved into birds is certainly debated, but there is certainly credible research backing this theory. The point being that, insofar as at least some dinosaurs may be directly related to modern birds, one could look at humans as coexisting with 'dinosaurs' in this regard. I realize that this last bit is not what you meant by coexisting with dinosaurs, I just thought I would throw that out there.
Because man never co-existed with dinosaurs.
Dinosaurs existed. We're not too sure if dragons did. In other words, Dragons exist in the "mind" Dinosaurs exist in the museums. Dragons also exist in literature and in scuplture. Not all dinosaurs are extinct.
No!
by a meteor
No
in the era of awesomeness when dinosaurs were alive.
people and dinosaurs never co-existed
No
no, but their ancestors did!
No.
No. They are extinct.
Yes. Birds are descended from theropod dinosaurs.