In Moodle 2.7 and above, in order to have a username, but also be able to use your email address as the username, you have to check a box.
Go to Site Administration / Plugins / Auththentication / Manage Authentication, and check the box labeled "Allow log-in via email."
The label is a bit confusing, since you are not actually logging in via email, but are logging in by using your email address as your username, even when you already have a username that is something else.
If this works as promised, you can instruct students that, if they forget their usernames, to use their email address instead.
Then they only have to remember which email address to use!