Have another go! Sometimes its a glitch.
Sometimes (not very good) hosting companies add a file to directories to tell which version of PHP to use – that could stop an update.
If no joy, have a look at FTP and see if any file permissions are not what they should be.
Files are 644, directories are 755 (https://stackoverflow.com/questions/18352682/correct-file-permissions-for-wordpress)
Try installing Sucuri – that will check for any weirdness.