This question is based on the shader in this answer. When I render the scene using plain glass for the material ball, it takes about 500 samples to converge. Here is the node setup and the result at 512 samples:
When I use the dispersion glass material, however, it produces a large amount of noise. Here is the node setup and the result at 2048 samples (4 times as many samples, which should be more than enough to make up for using 3 glass nodes).
Using mix shaders with 3x brighter glass produces the same result. The image does eventually converge after around a hundred thousand samples, but that's obviously not practical for real-world renders. Here is an overnight render at 65536 samples:
I can get a similar, but slightly different, result in far fewer samples by rendering 3 separate images and combining the color channels in an image editor:
But that leaves me with two questions: first, why does using the add shader produce so much more noise than expected? Second, why does it give a different result than a normal glass shader? Specifically, where does the black band at the bottom of the material ball come from?
Here is the .blend file, for reference:
All of this is observed in the material ball scene as of Blender 2.76. There are no lamps in the scene, and enabling/disabling multiple importance sampling of the world background doesn't have any significant effect on the noise. I haven't tested it in other scenes or other versions of Blender/Cycles.