You' ll probably say that Joomla can't have any connection to FBI? We also thought so and decided to share the experience which looks funny now, but it wasn't so funny once the story began...
Permanent login via 'Remember me' functionality is broken in Joomla if 'System - Cache' ('System - Page Cache' since 3.4) plugin is ordered before 'System - Remember Me' plugin.
Sometimes after update to Joomla 3.4 your sites can stop working if you are using native Joomla multilingual functionality.
How much time did you spend adding correct ALT and TITLE tags for images of your Joomla website? If more than 5 seconds - next info can be interesting for you.
Probably you faced a problem when using Joomla SEF URLs and once menu items aliases are changed or a site is moved to new domain or subfolder - all links to menu items are broken in your articles.