Annotated hexdump of the file attached, this took us a bit to decipher on IRC
Seems likely we should just use one or the other; I'd vote the smaller (but much cleaner) 16x16 image.
I see a lot of talk (http://chatlogs.musicbrainz.org/musicbrainz-devel/2012/2012-07/2012-07-22.html) about confusion between different images, but not what the actual problem is. The .ico format exists to store multiple sizes of an image so that whatever is using them can display the most appropriate size. It'd be nice if all resolutions were actually of the same image, so is that the problem that's trying to be solved?
The problem as I see it is primarily that we aren't actually using the larger image – none of the browsers we checked, as I recall, actually used it as a favicon. We'd see it when we opened it up manually in the browser (i.e., went to the URL of the favicon itself), or if we used some programs that (apparently) preferenced the larger image, but not otherwise. Hence removing the larger 32x32 image – it's barely used, outdated, and makes working with the file confusing.