Further to Chris' answer:
Yes, weather (or the equations describing it) is extremely sensitive to the initial conditions. The fact that the weather system contains phenomena at pretty much all time and space scales (have a look at figures 1.1 and 1.2 of these notes for some examples) does not make predictions any easier. Also, the "weather" (at various scales) is dependent on a whole bunch of other planetary systems (oceans, land-based processes, etc.) that come in with their own model and observational uncertainties:
Figure taken from Bauer et al., Nature, 2015 showing "physical processes of importance to weather prediction".
That said, significant progress has been made in the past few decades, and people in the operational weather forecasting community (at least the folks over at ECMWF, which are really at the top of their game when it comes to weather forecasting) refer to this graph that showcases these improvements:
The figure above is from Bauer et al., Nature, 2015 and shows a measure of forecast skill at three-, five-, seven- and ten-day ranges, computed over the extra-tropical northern (NH) and southern hemispheres (their caption).
What the figure shows is that forecast skill has been steadily increasing for the past 3 decades, with some major "bumps" in performance brought in by
- the advent of satellite (space-borne) measurements, which significantly narrowed the gap between the southern and northern hemispheres (the ground-based/sea observational network in the SH is much sparser than that of its northern counterpart).
- the introduction of a probabilistic (ensemble-based) weather forecasting framework
For example, the 10-day forecast quality is the same as that of a 7-day forecast made 30 years ago, so you can say we're up a day every 10 years :)
So I, for one, think that there is no hard physical barrier (boo, Lorenz!) that does not/will not allow us to predict the weather, say, 14 or 20 or 25 days in advance. It's just that we're lacking (1) sufficient knowledge of the weather system (2) computing power to simulate the weather at high-enough resolution (3) high-quality observational data to constrain existing 3D/4D-var and ensemble forecasting systems.