This is a common issue, if the system mailboxes are offline or not reachable. Technical background: login to ECP goes through OWA for authentication. OWA *always* requires a mailbox to sign-in. For Exchange Admins without mailboxes (not a mailbox user) or Exchange background tasks (e.g. mailbox move and migration), Exchange is silently utilizing one of the system mailboxes (arbitration mailboxes) for authentication.
So, if Exchange admin users do not have a mailbox (which is Microsoft best-practice for security reasons) AND the system mailboxes aren't available/reachable for some reason, the Exchange admins won't be able to login to ECP! This an Exchange architecture dependency only little known within Microsoft support ;-)