That's how it's supposed to work, Crow. "Remember Me" keeps you logged in, and not checking it will log you out, that prevents others from using your account when you're not on it.
My computer used to do that a lot, but it would log me out almost instantly. A lot of times I would sign in, go to post, and when I press the "post quick reply" at the bottom, it would take me back to the log in screen like I was never logged in.
I think it may have something to do with the browser you use because I switched mine and haven't had that sort of problem again.