No, humans and chimpanzees cannot have babies. And no, their DNA is not that similar.
The examples given at the end of the video explain why some animals can interbreed: they are all animals of the same kind… Horses and donkeys are both Equidae, Lions and Tigers are both Felidae.
Furthermore, all the evidence we have today for Neanderthals points to them being just modern humans. Their faces have been reconstructed by modern computer-assisted tech and they look just like us; they buried the dead, had tools, etc just like us; their childbirth was difficult just like ours (unlike previously postulated); they walked upright just like us; and much more. I mean, they are us, it’s just that people don’t want to admit it because it ruins their fairy tale of human evolution.
At the end of the day, Baraminology explains the data much better, as it’s clear we have different kinds of creatures, and within those kinds there’s nearly always the possibility to interbreed (and when there isn’t is because a creature’s genome has lost too much in the adaptation/selection process).
Thus, humans can only interbreed with humans because there’s only one humankind, as science has finally proved, too. To quote Robert Jastrow:
“For the scientist who has lived by his faith in the power of reason, the story ends like a bad dream. He has scaled the mountains of ignorance, he is about to conquer the highest peak; as he pulls himself over the final rock, he is greeted by a band of theologians who have been sitting there for centuries.”
Of course, we mustn’t take Jastrow out of context. Science is beautiful, is awesome, and would be more much more so, and even more productive, if it operated in the correct worldview. The same worldview that gave birth to it. To use the words of evolutionary anthropologist and science writer Loren Eiseley:
‘The philosophy of experimental science… began its discoveries and made use of its methods in the faith, not the knowledge, that it was dealing with a rational universe controlled by a creator who did not act upon whim nor interfere with the forces He had set in operation… science… owes its origins to an act of faith that the universe can be rationally interpreted, and that science today is sustained by that assumption.’