Approximate solutions are found computationally.

Non-trivial closed-form solutions are hopeless. Existence of solutions is a little less hopeless. Existence of solutions over small time horizons has been proven. This is all in three space dimensions.

There is a theorem stating that if global-time solutions exist and are unique, then computationally computed approximate solutions are 'good'. Moreover, they get better as you refine the computational domain.

Is finite element method good enough? Can you provide a concrete example where approximation is not good enough.

Special approximations called 'Galerkin approximations' converge.

Finite element methods are a special case of Galerkin methods. Finite element methods are good enough.

At this time NS solutions are all approximate, but there's a huge gulf between plausible-looking fluids (as in story), and fluids with pressure, velocity distributions accurate enough for analysis (to design an airplane). And within the latter there are further demarcations depending on the type of fluid analysis, where errors manifest as extremely subtle (pressure looks slightly high, vortices dissipate too quickly or too slowly) and judging the best algorithm for a problem becomes almost an art.