Is it possible for modern day humans to be part Neanderthal? Could they also share some of the same physical characteristics?

Its not a question of possibility, Its a matter of fact.

Modern humans, Homo sapiens came into existence from the integration of several early Hominid species. One of those being the Neanderthal. So its very true, and very real that most modern humans today have remnants of Neanderthal DNA as part of their genome. It was the inheritance of traits from this inter-species integration that most probably allowed for our species as it is now, to overcome adverse conditions in the past, such as Ice Ages, tool development, societal construct etc.

It is though Neanderthals were, although of similar intelligence, to other hominid species, were less advanced socially, and technologically, but also possessed a larger frame, were stronger and more hardy than its relatives of the time. So inheritance of those traits was definitely advantageous for Homo sapiens as a species,

It's fact, different amounts across the globe also, not the only ancient humanoid either,

Did you ever take a good look at Ron Pearlman's picture?

Sure