What does it mean to be "former PhD student"? I saw the term "former Ph.D. student" in some article -- does it suggests someone accepted in a PhD program but did not get his/her PhD degree eventually?
That would be a valid interpretation provided the statement is about oneself. More likely they attended for some period of time and left before finishing. There are many reasons for leaving, some personal and some academic. You can't draw conclusions about why they left from a simple statement. But, most likely, they have no intent or no possibility at this time of returning. 

However, as user JeffE notes, if it is made about another it is more likely to mean that the other person was once the writers student. In this case it is most likely that the person did finish the degree successfully, though it is possible not. 

A third possibility is that a person starts their study at University A, but completes it at B. Even in discussing themselves in a conversation about A, they might use such phrasing.