I want to know the name of an anime.?

I saw it on adult swim about six-nine years ago and the only part I remember was that some chick was in a coma-like state. Her friends are with her and are protecting her from some person. While she is unconcious, she has a dream that she befriends some person and the person wants to know her name. She then wakes up saying, "My name is ____!" She says her name, but I don't remember what her name was. I just started getting into anime and this popped in my head and it's killing me. I remembered it from about six-nine years ago, so the description may be wrong. If this sounds familiar in any way, please give me the name of the anime you think it may be from. Anything helps.