Most importantly, the Navier-Stokes equations are based on a [continuum assumption][1]. 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][2] 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)


  [1]: http://en.wikipedia.org/wiki/Fluid_mechanics#Continuum_hypothesis
  [2]: http://en.wikipedia.org/wiki/Knudsen_number