I have had issues with browsers that continue to cache the old css file even after clearing the cache and possibly you are facing this same issue ?
One solution is to add a version number after the file name that you can increment whenever you make changes as this will force the browser to reload the newly updated css and then cache it again
something like mystyle.css becomes mystyle.css?0001 mystyle.css?0002
On my website I decided to use 1 line of php to append the unix file creation time to my css filename so that everything was automatically updated instead of manually changing the value