Ok, good thinking!
Indeed it could be the backup completes but the email is not being send.
Only if the Backup Method setting is currently set to “Save Locally and Email” there could be recent backup files in the:
wp-content/uploads/ithemes-security/backups
folder on the server. If so it’s just the email that fails.
However the Database Backup module includes quite a lot of settings. Therefor it’s best to get the complete picture.
As of the 7.0.2 release the plugin includes a debug page which can be enabled by adding the following line to the wp-config.php file:
define('ITSEC_DEBUG', true);
Note it’s ITSEC_DEBUG in single quotes. Perhaps better not to copy/paste. Normally I would say copy/paste but in this case type it in manually to prevent errors. Do double check …
Also make sure to add the line in the correct location in the wp-config.php file. It must be added before the line below:
/* That's all, stop editing! Happy blogging. */
Once properly added to the wp-config.php file navigate to the Security/Debug menu option and then under the Settings section select “backup” and finally click on the Load button.
Post ALL the information displayed on the Debug page.