What can be causes that script in /etc/cron.monthly/weekly/daily/hourly folder does not do its job/not executed/not run?

I think this may help:

check that script has right permission (-rw------- root root)

# stat /path/to/script

now change to proper permission for Linux cron

chmod 600 /path/to/script

check that there is new line at the end of script

command on the last line
# this is the end
check that cron service is running (pid exist)

# pgrep cron

try to add this line to the beginning of the script:

#!/bin/sh

try to use full pathes for commands in the script

like not "find parametershere", but "/usr/bin/find parametershere" to discover full path, do example. "which find" command or "which grep" etc.

check that user that should run the script is defined

if the script is launched from an file in /etc/cron.d/ directory, then try to add "root" into the file, ie.:
0 */1 * * * root /usr/sbin/tmpwatch -am 1 /tmp/nginx_client