In some QFT texts one writes down the number operator $N$ for free theories, such that when acting on an $n$-particle state $|n\rangle$ we have
In free theories this is a conserved quantity. However I have never seen this quantity derived by using Noether's theorem, i.e. as a consequence of the invariance of the action under some transformation of the fields or coordinates.
Is it possible to derive the number operator via Noether's theorem? If not, is it possible for a theory to have more conserved quantities than just those accessible to Noether's theorem?