You can always argue that you can predict weather for next milli or micro or femto seconds, why stop at weeks?
The reason that I've heard for stopping at weeks is that that is the time frame for perturbations to work their way up from the quantum scale to the macro scale. We cannot, even in principle, measure what is happening everywhere on the quantum scale.
Sometimes these smaller perturbations are called "sub-grid phenomena". I think practitioners think of them as, say, pressure waves or flows which average to zero when observed on a 10km x 10km grid. But it is possible that their ultimate origin is on a much finer grid ;-)
Another problem, separate from these effects, is getting closure on the variables in the model. Things like evaporation from soils and vegetation, for example, which ECMWF is trying to include in their models. You put them in the weather model to improve accuracy, and all the sudden you need a time-dependent model for soil and vegetation water content. And also, sensors to satisfy the boundary conditions for your model (e.g., to estimate plant vegetation type).
The reason that I've heard for stopping at weeks is that that is the time frame for perturbations to work their way up from the quantum scale to the macro scale. We cannot, even in principle, measure what is happening everywhere on the quantum scale.
Of course as soon as you merge quantum mechanics and chaos theory, life gets very, very weird. See http://www.iqc.ca/publications/tutorials/chaos.pdf for more.