No and i'll tell you why with 3 reasons.
First, we don't have the technology. We have tried to make simple life in a lab but fail, we think life came from an ancient earth soup, maybe a strike of lightning and whala abra cadabra, but truth is we do not know and have not been able to replicate.
Secondly, our understanding of genes and how these create biological changes is very limited. To give you an example, a lot of the animals we try to clone with existing material tend to die at birth. I am sure you've heard a lot of talk about cloning mammoths, while we do have their genes the cold truth is we do not have the science to clone a mammoth.
Thirdly and this is the major road block that we can't overcome regardless of how our science advances; we do not know ancient climate and we can't recreate this. There is a lot we do not know about ancient environmental pressures that caused evolution of certain species. While geologists can get a decent scope on ancient weather and atmosphere composition, they only get a very general picture with rocks. Evolution is also driven by competition between species, and sadly most living beings do not fossilize, and even those that do simply give us very little information on the actual living being, like its soft tissue, behavior. DNA sadly does not fossilize very well, the oldest partial dna that we have is a few million years old. We will never have nonavian dinosaur dna.
With that said, I do think eventually we can make our own animals, including our own fake nonavian dinosaurs, like the infamous chickenosaurus.