Would help to know more about setup... but given the information shared ...
Is Moodle set to run automated backups? Larger courses (like digital media could grow to 20,30,40,60 Gig range in a hurry if there are many assignments). Larger the courses, longer for the cron job. Might set that to manual - which means autobackups can only occur via command line.
Cron jobs empty the 'trash' - ie, removes files no longer linked anywhere ... so if one is running cron only once during a day the trash could be filling up. Look at size of moodledata/trashdir. By default, think trash is auto-emptied every 4 days.
Anyone been working on themes and forgot to turn off the 'designer mode'?
Site have many forums to which users are subscribed? Much traffic on those? Is the mail working? Mail could be bouncing back to the Moodle server - and depending upon how the MTA/Mail is setup, mailque on the Moodle server might have tons of messages that it is trying to re-send.
Can run the cron via browser ... it will show what it's doing and how much memory it's using up ... as well if something fails.
It could be other factors as well ... because of the 'un-responsive' bit ... is the DB able to handle the number of connections? is Apache able to handle the number of connections? What's traffic pattern like? Is there any 'prime time' for students? If autobackups set for same time as 'prime time' ... is there any wonder?
Is there any opcache installed? Has it become fragmented?
Just a few off top of head ... others might be able to 'read more between the lines' than I. ;)
'spirit of sharing', Ken