Not only is it possible, the field of stuttering was started by speech paths who stutter? I so wish when I was younger that I had a speech pathologist who stuttered. No one can understand how difficult it is to not say your own name, to be laughed out while ordering or relate to the fear of talking.
I never thought when I was younger I would be able to be so grateful to have a trait that has enabled me to help so many others. I am grateful that I can talk to children, teens and adults who stutter and say "I know exactly what you are feeling" and help ia available. I can model what I teach and show that it is successful. I can also show that every once in a while I stutter and the world won't all apart. My hat goes off to all the speech pathologists who stutter. I recommend listening to www.stuttertalk.com, which is lead by Peter Reitzes, a SLP who stutters and Eric Jackson, a grad student who stutters. It is a great inspiration.
I also recommend you follow my website as I will be putting before and after video clips of myself. It is worth listening to. I have overcome fear and severe stuttering.