The most likely cause for your apartment mailbox not working correctly is a conflict with older expansion pack buyable mailbox from Sim a Little Dream a Lot
Look in your Collections folder for a file called 0x6e870b24.package and in your Downloads folder for a file called SEDD_BuyMailboxBV.package and if you find either one, delete them. They are older versions of a very popular buyable mailbox often included in downloaded lots so you may have them and not even be aware of it, but they do conflict with the new Apartment Life mailbox so they must be removed.
There is a new version of the buyable mailbox already available especially for Apartment Life at http://daydream.etowns.net for those times you accidentally delete your mailbox or download a lot with the mailbox missing. Anytime you experience problems with your mailbox after downloading lots in the future, remember to re-check for the two files mentioned above and make sure you do not have them. The only safe buyable mailbox file to have in Apartment Life is SEDD_ALmailbox.package, note the AL designation.
If you continue to experience mailbox problems, even with the updated buyable mailbox, especially if your problem seems connected to Night Life date items being delivered, your problem may be addressed in this thread.