To get notified when a cron script fails you can tell cron to send any errors to you by email.
It already does this by default, but the emails are sent to the Unix account's mailbox:
cat /var/spool/mail/<username> | less
To use a different email address use the MAILTO setting as described here:
MAILTOfirstname.lastname@example.org 0 */2 * * * /bin/backup.sh > /dev/null
Cron will now send an email whenever the script (/bin/backup.sh) prints something to stderr.
If the script is printing to stderr without it failing try Cronic:
0 */2 * * * cronic /bin/backup.sh
You can of course also try to fix the script by not printing to STDERR unless it's really an error.
Check the mail log to see if it's working:
sudo tail -f /var/log/mail.log
If you're email server is rejecting emails sent by cron you might need to set the MAILFROM variable. However, only some cron version supports MAILFROM.