Till recently when Google Apps Premier email had a maximum storage limit of 25 GB, my email my.name@my-domain.com got full, and I was forced to purchase and make another username for myself name.my@my-domain.com. Very recently Google has removed that restriction and now any amount of storage (upto 16 TB) purchased, will be shared across Google Drive, Google+ Photos, and now Gmail.
Therefore, due to inconvenience and expense occurring in maintaining two user accounts for a single person, I have decided to dump my second email and transfer all the emails (in Inbox, Archived, and in other labels) to be copied or moved to my first email, with label structure intact.
Drive is not an issue since the ownership of entire Drive content could easily be transferred through a very easy way available in Google Admin Consol. It is the email which is a real pain and about which Google seems to have not done or said something about.
Best Answer
As of sometime in 2014 you can migrate emails using the Data Migration tool in the Admin console. This is meant for transferring data between domains, but you can use it to transfer data within your domain as well.
Before you start, you may want to log in as the source user and apply a label to all of their emails. Otherwise, they will just be thrown in the target inbox with just their existing labels, which can be hard to sort through. While you're there, grab an application-specific password if you have two-step verification turned on.