For a point-sampled image, blurring and thresholding it is roughly equivalent to downsampling the image and then upsampling it to the same resolution. Similarly, a fractal synthesis algorithm that works on the scale/add principle can be stopped at fewer levels of scaling and adding to get an image that's very approximately equal to a lower-resolution image that's upscaled (if that makes any sense).

For something like a basic fBm, the number of octaves of noise required to generate an image without aliasing (that is, that just exactly fills the surface with maximum detail) is approximately log2(imageresolution). That is, for a function with 1/(2^n) scaling, having more than the above number of octaves won't generate meaningful new information, while having a lesser number of octaves will generate versions of the surface that have the same information content as lower-resolution images. The above information is subject to the basic function, of course; the classic cubic-spline-based Perlin function gives nice a smooth lower-resolution images that look an awful lot like a Gaussian-blurred and thresholded image.