I recently added a second exchange account into a network user's Outlook 2010. Since then, every time that user logs in, Outlook asks for one of the two exchange account passwords. It switches every time between the two. Every time this happens, we enter the password and make sure that the "Remember this password" option is checked, but it still always pops up.
I have the same setup in my own Outlook (with two other exchange mailboxes) and it works just fine -- no constant password entry requirement.
I've compared the settings between my outlook and his. The only difference I noticed was that both of my exchange mailboxes appeared as "email@example.com," whereas one of his appears as "Mailbox - User's Name" and the other as "firstname.lastname@example.org."