Yes, bullying can cause PTSD (post traumatic stress disorder). Bullying is classified as assaultive violence, and assaultive violence is associated with the highest risk of developing PTSD (Breslau et al., 1998). Bullying causes PTSD as the victim is scared for their life from the aggression of the perpetrator. And the emotional abuse can be just as damaging (or more so) as the physical abuse.
Thus, schools and workplaces should have a zero-tolerance policy towards bullies.
Reference:
Trauma and posttraumatic stress disorder in the community: the 1996 Detroit Area Survey of Trauma. Breslau N, Kessler RC, Chilcoat HD, Schultz LR, Davis GC, Andreski P. Arch Gen Psychiatry. 1998 Jul;55(7):626-32.