Consider a standard volume of $1\textrm{ m}^3$ of air. This contains on the order of $10^{25}$ molecules of O2 and N2.
If you needed to simulate or explain the physics occurring in that volume of air, would you want to model $10^{25}$ molecules and all the interactions between them or, say, 100x100x100 cells based on the Navier-Stokes equations?
Theoretically, it is possible to simulate every fluid flow ever by tracking every single molecule. But direct simulation of turbulence using the Eulerian Navier-Stokes equations requires $Re^{9/4}$ grid points and is thus totally impractical for Reynolds numbers larger than a few thousand. So simulating something with $10^{25}$ things to track is completely impossible.
Most importantly, the Navier-Stokes equations are based on a continuum assumption. This means that you should be able to view the fluid as having properties like density and velocity at infinitely small points. If you look at e.g. liquid flows in nanochannels or gas flows in microchannels you could be in a regime where this assumption breaks down. As far as I know there is no hard limit for the continuum assumption, but the Knudsen number is a useful indicator.
Additionally there is, as @ShuchangZhang mentioned, an assumption on the nature of the stress in the fluid. Although I am not sure whether you could really call this an assumption or whether it should be considered a theory (like the NS equations itself).
The strongest assumptions are typically not in the Navier-Stokes equations themselves, but rather in the boundary conditions that should be applied in order to solve them. To give an example, whether the no-slip boundary condition (fluid velocity at the wall equals wall velocity) or the navier slip boundary condition (fluid velocity equals a scaled velocity gradient at the wall) has been a much debated subject, in particular for hydrophobic surfaces (see e.g. Phys. Rev. Lett. 94, 056102 (2005) and references therein and thereto)
Best Answer
Classification into parabolic, elliptic and hyperbolic equations applies to the second order linear partial differential equations with constant coefficients. That is, it applies to the equations that are:
Thus one particle non-relativistic time-dependent Schrödinger equation with no external potential or magnetic field can be classified as parabolic (although with complex coefficients - its real-coefficients equivalent is the diffusion equation.) In time-independent case in more than one dimension, Schrödinger equation is elliptic.
Navier-Stokes equation is non-linear, and hence does not fit this classification.