More information for you ...
I tested this several times, and when it did finally happen again, it was doing this:
Logged in (automatically logged me in with my admin account, as "remember me" was previously selected) and impersonated my own user mail account.
I left both tabs open. Truthfully, I forgot about them and ran some errands. Both tabs were open and idle for probably 2hrs before I got back to my desk.
I closed down the browser entirely and re-opened it. When I navigated to the mail server, it automatically logged me in as my user account and not the admin account.
So there has to be a timeout or something happening where one session expires before the other, and on expiration, the user account is overwriting the remember me data.