Can static electricity and electric current occur simultaneously in the same object?
For an electric current in a conductor, we require an electric field in it to cause the charges to drift. But, in a static situation where the conductor is charged with charges at it's surface, the net electric field inside it must be zero.
The above result can be proved using Gauss Law. Thus, the two situations cannot occur simultaneously and as such it is not possible to have electric currents and static charges in region of a conductor.