By creating the following script:
#!/bin/bash
file=/tmp/test$RANDOM
touch "$file"
sleep 600
rm "$file"
And running it every minute in my crontab.
guido@solid:~$ crontab -l
* * * * * /home/guido/cmd.sh
I can see that there are many instances of cmd.sh running by:
guido@solid:~$ ls /tmp/test*
/tmp/test7822 /tmp/test12278 /tmp/test16118 /tmp/test25642
/tmp/test11429 /tmp/test15958 /tmp/test18172 /tmp/test26104
guido@solid:~$ ps -e |grep cmd.sh
26005 ? 00:00:00 cmd.sh
26454 ? 00:00:00 cmd.sh
26999 ? 00:00:00 cmd.sh
27450 ? 00:00:00 cmd.sh
27895 ? 00:00:00 cmd.sh
28667 ? 00:00:00 cmd.sh
29220 ? 00:00:00 cmd.sh
29727 ? 00:00:00 cmd.sh
30232 ? 00:00:00 cmd.sh
This means, your mailing job is being ran multiple times, which may lead to some problems. Either up the interval until you're sure that there's no overlap, or change the job to take a lock so there are no issues.
As to why there's no indication on your mail being sent, I can't help without more info.