In my case, at first I didn’t get any errors in the cron logs in <web-root>/var/log. When running the scripts by hand as the website user I got the not so informing “Cron readiness check failed” in the <web-root>/var/log/update.cron.log.
The reason the “Cron readiness failed” was actually due to some files in the file and directory structure in the web root that can not be written by the cron update script. In my case I used the AWstats package to create website statistics placed in a directory called /stats. This directory and it’s file was not writeable by the cron script, causing it to fail, even though the directory /stats and it’s files are not a part of Magento.
I discovered this by investigating the content of the file var/.update_cronjob_status. In this file you can find important information on why the cron update script is failing.