Is it possible for a PowerShell script to monitor changes to a file within a share and run another program if a single users changes x amount of files within a given time period?

Yes it is.

Thank you, are there any sample codes available to test?

Here are a few: https://gallery.technet.microsoft.com/scriptcenter/site/search?query=filesystemwatcher&undefined=undefined&ac=3

Please refer to this informative technet article that seems to be an appropriate place in order to resolve your concern in depth : http://blogs.technet.com/b/mspfe/archive/2013/08/27/auditing-file-access-on-file-servers.aspx

Here is another PDF guide that covers step-wise instructions to enable auditing on file server and track every critical changes into real time : http://www.lepide.com/guide/enable-file-folder-access-auditing.pdf