I am not talking about NP-indeterminate class because those problems have to be shown to not exist either in P or NP-complete class and existence of such problems proves P!=NP. I am interested to know if we are half-way there i.e. problems that have been proven to be not NP-complete but are in NP but we don't know if they are in P as well.
What about such problems whose deterministic polynomial solution was found only recently? Was primality testing such a problem?
I would appreciate if the answers are given for someone like me who only has a high level understanding of computational complexity theory.