How much further can weather forecasting models go before the limits of Chaos Theory set in?
There are (at least) two factors involved here, before we even get to questions of chaos theory.  First, thanks to weather satellites and other sensor improvements,  we can measure the initial conditions much more accurately.  (One can do a halfway decent 3-day forecast for the West Coast just by eyeballing the GOES West images.)

Second, we have greatly increased computing power.  Forecast models (simplisticaly) work by dividing the world, or part of it, into a 3D grid, applying a set of differential equations in the grid, propagating the changes to neighbors, and then stepping forward in time to repeat the process. The finer you can make the grid, the more accurate the forecast can be.  However, if you cut the size of a cell in half, you need to do 8 times as much computing.  And if it takes longer to do your computation than the actual weather, it's not much use as a forecast :-)