I was getting the following error in my logs whenever my server would attempt to e-mail me my daily backup file (8mb):

postfix/postdrop[21071]: warning: uid=1001: File too large Mar 28 00:00:11 xxxxx postfix/sendmail[21070]: fatal: xxxxx(1001): message file too big

To solve this, the maximum outgoing message size needs to be increased. Run the following command:

nano /etc/postfix/main.cf

Add the following entry to the end of the file:

message_size_limit = 20480000

Run the following command to restart postfix:

/etc/init.d/postfix restart

After doing this, the server was once again able to send out my backups.