Okay, I realize that this is a simple one but it caught me by surprise nonetheless. Most of our scheduled scripts are placed in the /etc/cron.daily, /etc/cron.weekly and /etc/cron.monthly directories. After migrating some of these scripts to another server it turned out they were not being executed. After some research it came up that run-parts (or at least some implementations of it) was just skipping the files that had . in it. Our CentOS server was executing them perfectly but the Debian server just skipped them. The man page mentions it by omission:

If  the  –lsbsysinit  option is not given then the names must consist entirely of upper and lower case letters, digits, under-scores, and hyphens.

To check what scripts will be executed call run-parts with the –test option.