When using the web interface to Dartmouth email (BWA), to login to a second Dartmouth email account at the same time you are logged into the first account, you will need to use a different browser, or a private browsing tab.
This is because once you log in to a Dartmouth email account via a web browser, even if you open a new tab, your credential are being carried to the new tab. To open a separate Dartmouth email account, you need to launch a different browser, or open a private browsing tab, to be able to enter a different set of log in credentials.