How much further can weather forecasting models go before the limits of Chaos Theory set in? Refer to a report here that mentions the huge advancement in weather forecasting models since the 1980s:


  How much better? “A modern five-day forecast is as accurate as a
  one-day forecast was in 1980,” says a new paper, published last week
  in the journal Science. “Useful forecasts now reach nine to 10 days
  into the future.”


But weather forecasting cannot be indefinitely accurate, one fundamental limitation is Chaos Theory:


  Chaos: When the present determines the future, but the approximate
  present does not approximately determine the future. (Wikipedia)


I used to think some ten or twenty years ago that the accuracy level of 3 day weather forecasts was due to Chaos Theory, but apparently it wasn't. 

Are there studies where weather forecasting models are subjected to sensitivities of initial conditions, investigating how Chaos Theory puts an upper limit on the accuracies of such models? Also, based on current models, 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 :-)