We are moving into weird heterogeneous bugs territory. A lot of stuff that has wasted my time last year was caused by interaction of technology stacks and not by pure logical errors. Or error localized in even one package.
I had problem with premature socket close on the response with just the wrong setup for all of nginx, haproxy, slim and php 5.4 ... lets pure ops or pure dev figure that out.
I rely on the ops guys to keep vanilla installs steady and to keep everything patched, but there's very little way that they can get insight into application problems (especially in the framework du jour) that express themselves at some other location in the stack. I just have to know the whole thing.
I on the other hand have no admin rights (only read access) but I know enough of out network and ops structure to be able to make an educated guess when something is behaving strange.
So if it hits the fan they could probably patch an emergency bug and I could fire up some VM instance. It is not about wearing a lot of hats, but being able quickly to put on a new one if needed.