Can you call yourself a vet and get benefits if you don't complete basic training?

I have an ex that went to basic the same time I did but she didn't finish. As the Mother of my child we keep in regular contact with one another despite the differences that we may have. She asked if she would be considered a Veteran and get Benefits since she did join the Army. I believe if I recall correctly she was given a General discharge for failure to adapt. I told her not only could she not call herself a Vet as she didn't even complete a month of Basic, but she didn't qualify for any benefits either. She was insistent that she was...Am I wrong?