I consider myself a good database Developer/Admin, but you wouldn't want me messing with your website configuration. Where I work now, we have a dedicated DevOps team. They do things like IIS configuration, domain account administration, instrumentation and automation for deployments, etc. Essentially they perform an assortment of roles and tasks across the enterprise that in some organizations a developer or database admin might have been expected to do occasionally, except a DevOps specialist takes it to the next level professionally. But, of course, everyone in IT is a link in the overall DevOps chain; it's not just a department. Even a small company with a handful of IT employees can apply the principles of DevOps.
I see this happening as companies look to adopt more of a DevOps mentality. A team focuses, and slowly trains others. Eventually many departments are involved, in their way.