The new standard email client with Android 4, doesn't seem to keep local copies of emails once they are deleted off the server.
Honeycomb did. The email was downloaded to the tablet, and kept even after the email was deleted from the server.
I'm using an Acer A500 with Android version 4.0.3 (kernal 220.127.116.11+).
Email set up is 'normal'/standard POP3 account. Setting is Delete email from server: Never What am I doing wrong?