Just to add to this, I got the problem resolved for the user reporting the problem at my company.
Exchange 2010 has a hard limit of 10 ActiveSync partnerships. If you attempt to add more, they won't work. This isn't a problem for most, but for people with multiple devices and people who frequently upgrade / replace their devices, this is an issue, as it proved to be in this case.
Deleting an account from a device does NOT delete the partnership on the Exchange Server. The Exchange Administrator needs to go to your account on the server under Manage Mobile Phone and remove the partnership for any device(s) you're no longer using. Note that the Outlook app counts as a separate device, so if you were to set up both mail.app and Outlook.app, that counts as 2 partnerships, even though they are on the same device.
Hopefully, this will help point someone in the right direction.