So is with the chaotic systems: we know for sure that approximating them is just a matter of boundary conditions: that is - the more you know about the starting points, the better you approximate the real world.

I mean, there might be a case where we will finally get to the point where we are just "good enough" at measuring the boundary conditions to predict weather for the next year, but it has got nothing to do with obtaining the knowledge about chaotic systems themselves.