Two counterpoints:

1) What you've written, where H is independent of t is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics. Even with time-dependent hamiltonians we have decent tools for talking about the solutions (Dyson series).

2) Of course for certain values of H, (esp. in continuous space) you can contrive ways of making the eigenvalue problem hard, but you don't have to go as far as quantum mechanics to find difficulty. Just take three bodies under newton's gravitation.

Yes, the quantum n-body problem is exponentially harder in n, but that's a fundamentally different type of "hardness" than the hardness of Navier-Stokes.