I have been doing neck bridges for 32 years, and I have not had any problems, in fact I think the neck bridges are what have so far saved me from neck problems after 32 years of wrestling.
I do agree with Theefaulted, you do have to do them properly.
I actually see more BJJ guys with chronic neck problems, than wrestlers, and I attribute it to the fact that wrestlers are actively exercising our neck. Many BJJ practitioners neglect neck exercises because they feel they do not need them,. because they do not bridge over their head. A strong neck makes your vertebrae more stable.