Curse of Dimensionality
Our intuition does not serve well in high dimensional spaces. Hence there are few issues with using nearest neighbor methods on high dimensional data. Firstly, the methods that involve capturing a fixed neighborhood around the points gives a high variance for the fit. Secondly, if you relax the fixed neighborhood criterion and try to capture a specific number of neighbors, the methods are no longer local. Hence it pays to think through these issues on whatever dataset you are working on. You might expect low variance fit but the curse of dimensionality shows up and you get a high variance fit.