I had a similar issue, but not an NPE, rather the family were from India and any records that might shed light on the matter will be in a church there somewhere.
My mother's mother's mother was born in India in 1899/1900. And that's all we knew. From quite extensive cataloging of DNA matches I was able to determine who one of the parents are. This was done by finding DNA matches triangulating seven different matches. However, this has only been possible due to the sheer number of tests I have - 10. With enough tests you can rule the siblings out.
So let's say this theoretical great-grandfather had four siblings. Really you'd need a test from each of their descendants.
I have a similar situation with another match. My aunt has a full first cousin match - doesn't fit in any other way. She was born out of wedlock. So all I can say is that it was not my aunt's father who was the father - it was one of his four brothers. I can make a good guess as to who the father is, but unless one of the children of the other brothers takes a test, the identity can't be known.