I should rephrase this. User have to issues accessing Sharepoint, however depending on how they access the site, they may or may not be able to checkout documents.
We have a Server 2008 R2 install with Sharepoint 2013 in a 2008R2 AD domain. Domain users can access the site by either http://oursharepoint.domain.pri or http://oursharepoint.
When accessing by the short name, there are no issues. Users can check out and edit documents. Both addresses resolve the same. The workstations are on the same domain. When the FQDN is used, I am usually challenged for credentials all of a sudden. After submitting the credentials I can continue, but when I try to check out a document for editing it says This Document Cannot Be Checked out... you don't have permission.
Either case the user can check out a read-only document. Why would this make a difference to intranet users who are already logged into the domain?