I have seen similar behaviour when the website is not being redirected consistently.
In Joomla 3.x, you probably shouldn't be using the livesite parameter in the configuration.php file but if you are, make sure it's consistent with any redirections in your .htaccess file or anywhere else. For example, if you are redirecting the non-www version of the website to the www version of the website, make sure the livesite parameter (if you are using it) is set to the non-www version of the website.
Back up your .htaccess file and temporarily copy the contents of htaccess.txt to .htaccess to reset your .htaccess file back to defaults to see if something in .htaccess is causing the issue.