Yes its possible

Yes it is if it's anything like my phone having the app sometimes shows that you're online all the time

I do believe that it's quite possible.

I believe that I have experienced the same thing.

I was told that the reason someone could still message me (although my profile was hidden) was because they saved me as a favorite, so my profile was still visible to them. So, before I deleted my profile (just in case it wouldn't really be deleted), I deleted everyone who saved me as a favorite so that there was no excuse in being able to receive a message. I also deleted all my pictures a whole ago, along with my"description" because I've heard that POF steal profiles and pictures and creates new profiles all over the country with your pictures, etc. Never again will I go on that site.

POF states that the cookies follows you and so although not on their site it will show online while on another site or not logged off other site. BIG BROTHER COOKIE

I did an experiment. I created a phony profile to check and see when I hit the invisible option if my real profile would stay hidden. It didn't. Also, someone wrote to me while I was in invisible mode. There is an option that is on the settings page that gives you the option of being invisible. Also, if someone is invisible and you type in their user name in username search, their profile shows up.

Their crappy phone app does this

I've never encountered that situation. If I messaged someone that says they are on line, they usually answer back pretty quickly. If they don't answer back, I take that as a lack of interest and move on.

They are online and just don't want to admit it. They claim they never logged off. Excuses!