This question can be asked about any chaotic dynamical system, but hydrodynamics of the atmosphere makes it more concrete. Arnold describes his 1966 result as follows:
I have calculated the curvature of this group [diffeomorphism group in hydrodynamics] and even used it to show that weather prediction is impossible for periods longer than two weeks. In a month you lose 3 digits in the prediction, just because of the curvature. This instability is not the Euler instability, it’s not describing a chaotic attractor of Euler equations – but it comes from the same line of ideas.
How final are the two weeks? One could imagine collecting data of greater precision and getting a meaningful longer term forecast. But there seems to be a theoretical limit to increasing precision, as with the diffraction limit to optical resolution. At a too fine enough precision, positions and momenta can not both be specified even theoretically, and the classical description breaks down. Quantum effects are usually negligible at classical scales, but does this apply to chaotic classical systems? In them initial discrepancies quickly magnify. Does this mean that quantum effects become classically relevant and long-term prediction of such systems is impossible in principle? Is there a theoretical time limit on weather forecasts for example?
Practical limits are discussed in How to calculate the upper limit on the number of days weather can be forecast reliably? Apparently, 15 days come up as well.