**Redirection Consistency**

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 www version of the website.

**.htaccess**

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.

**Joomla Update and Check for Malware**

You should also upgrade to the latest version of Joomla as Joomla 3.4.1 has some known vulnerabilities. I'd be inclined to scan the website using myjoomla.com or similar in case the website has been compromised and this is the reason you are having intermittent login issues.

**Third Party Extensions**

What third party extensions are enabled on the website? Try temporarily disabling these one by one to check if a third party extension is responsible. Try updating all third party extensions to the latest versions as this may also help fix the issue.

**Session Table**

The `[prefix]_session` table is frequently updated in Joomla and can occasionally become corrupt. Run a repair on the table using phpMyAdmin or Akeeba Admin Tools or similar.

**Front-end / Back-end**

Are the back-end and front-end logins both affected or only one of these? This information may help narrow down the issue.