Yes, this has been happening to me as well (I've been wondering if it was just me or was happening to others as well).
Actually, it just happened as I was trying to reply to this thread.
I too am using Private Browsing
I'll try checking the Remember Me box and see if that helps, although will that actually remember my login ID and fill it in auotmatically next time, even in Private mode?

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Okay, so did some testing.
Checking Remember Me does seem to keep it from logging me out, and no, it appears it doesn't actually remember and autofill the login ID the next time you login if you are using the browser in Private mode (at least not for Firefox).