Google will remove search results when a court orders them to do so or when the search result can infect a computer with viruses and malware. We know that.
I spotted an interesting and unique thread at Google Webmaster Help thread where one individual (won't use the name but you can easily find out in the thread) had a court order Google to remove search results. The search results were about a sexual harassment case between her and a guy from ten years ago. She said, when you Google her name, it comes up and thus is hurting her chances of finding employment.
So what is the issue? She said one of the results is still there, but partially removed, making it worse. She said:
One of the links is gone, but this one remains, and now it is untitled, but the snippet reads about how I sued someone for sexual harrassment, with my name and his. Before, at least, that information was not in the snippet.
Here is a picture (again, I don't want this article to rank for her name, so I am not using it in the text):
[screen shot removed on behalf of this person]
You can see the third result is untitled, meaning, it is likely partially removed. If you want to see the full court case, either Google her name or go to the URL she provided at the thread.
My only good suggestion is take this back to court. I doubt anyone in the forums can help her.
Someone said try this form, but she said she did and they told her to go to the owner of the web site.
I have never seen someone with a court order come to the forums asking for the court order to be upheld completely. So that is why I found this interesting.
Forum discussion at Google Webmaster Help.
Update: Please see the comments below for more details and errors.