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As far as I know, once you login, you stay logged in until you logout or close the tab/window. There's no setting to logout after a specified period of "idle" time and I don't believe the site has such a setting programmed (because I've been logged in and idle for a LONG time (I mean a few hours) and still been connected when I check). I do notice this happens most often when composing a long post (so it it possible that being idle while in that screen does cause a logout - I'm not certain nor do I have any idea how long it takes if it even exists), but if so, there may be some setting there that only the Administrators can control - if even they can. We can do nothing about it.
This has happened to me (not logging out exactly, but loosing connection with the website which I suppose amounts to the same thing). I don't know why, but it happens (rarely, but sometimes). And not just here, but on other websites (I suspect a brief glitch in the network connection that's enough to log you off the site and no, I'm not sure how to fix it and haven't tried as it doesn't occur that often).
If you get such a message again, copy your message before doing anything else. Then when you log back in, you can start a new thread and just copy the contents of the old one into it and the problem will be minimal. Sometimes on a long message, I'll make periodic copies as I go along so in case that happens, I don't have to restart from scratch.
"Remember me" will help you log back in more quickly (automatically), but you will still lose the contents of the message you were composing - it is not saved from login to login (automatic or manual).
I hope this helps.