ANSI .PST files were limited to 2GB, but unless you have very old PSTs you will be using Unicode (since Outlook 2003).
Unicode PSTs can be up to 50GB by default in Outlook 2010 and above (20GB in 2007), and this can, in theory be increased to the technical limit of 4194304GB (which is 4096TB or 4PB) via registry changes, but probably not advisable or necessary!
https://support.microsoft.com/en-gb/kb/832925
To determine Unicode or ANSI:
Am I in ANSI or UNICODE format? - MSOutlook.info
Notwithstanding the ability to have humungous (technical term) .PSTs, it's probably better from a performance, resilience, backup and usability perspective to have separate PSTs for archived messages and also for sorting mails - instead of folders within default profile PST, create dedicated PSTs for the larger folders and move messages in there.
His suggestion was to use the "restart" option at least three times, not to "shut down" and then re-boot.
My suggestion would be to change your IT man.
SD