If your GGM was born in a Christian family, and converted to Judaism, she it totally, thoroughly Jewish according to traditional Jewish law, and DNA knows and shows nothing about it.

If your GGM was born to a Jewish father and a non-coverted gentile woman, she is not considered Jewish according to traditional Jewish law -- no matter how much "Jewish" DNA she has. 

Personally, I find it rather jarring to hear people talk of Judaism as though it were a DNA issue.  We know who thought being a Jew is a racial issue, and it was not the ancestors of those of us who consider themselves Jewish.

DNA can indicate family relationships, or belonging to certain geographical group.   I know Jews from  Marrocco, Kurdistan, Ethiopia - since when is being a Jew related to having come from Central Eastern Europe?

So, some questions you should ask yourself about your GGM, that have nothing to do with DNA:

1. Why do some descendants think she was Jewish, and some not?
2. Was there any doubt if her own mother was Jewish? Was it clear she wasn't?
3. Was she a Jewish woman married to a Gentile, bringing her children up as Christians?
4. Was she a gentile who converted and married a Jewish man? Did someone want to cast doubt on her conversion?
5. Did anyone want to actively hide / reject the Judaism of their ancestors and family -- and if so why?  Some did for fear.  Others in the hope of gaining social advantage / professional status -- etc.

--
Odeda Zlotnick
Jerusalem, Israel.