I dislike the FB chat feature. I try to keep online so my friends can pop a quick hello, but I get many "can you click this link and help me with this war/monster/job/etc", which I find very intrusive. Unfortunately FB chat and Flamingvixen do not get along. So while I do most of my gaming from chrome, I check my groups/status updates/etc from Flamingvixen.
The problem with sending me a chat thingy? I may just not be here or on that tab to see it. I may be reading eHell, reading up on some WoW stuff, the news, or even playing WoW in full screen. The kids could be on my PC, or a cat could have decided to perch on my mousepad.
As other posters have mentioned, sometimes FB chat just drops messages, or it shows that you have dropped offline when you really haven't. It's kinda also like any instant messenger, sometimes you just get called away, or focused on something else, and totally forget you were having a conversation with someone.