Aliasing is vary easy to notice even on a 4k screen. Look at the second immage it's got way to much white and you can easily have the same issues at 4k or 8k.

http://en.wikipedia.org/wiki/Aliasing

In the end increasing resolution does help in most cases, but with the right fractal pattern there is no 'safe' resolution.

PS: It's basically the same reason that QuickSort is a O(n^2) sorting method in the worst case. Pick the wrong data and your assumptions fall apart.