Wordpress - Can't log in: "ERROR: Cookies are blocked or not supported by your browser. You must enable cookies to use WordPress."

Add below line to your wp-config.php before /* That's all, stop editing!...*/

 define('COOKIE_DOMAIN', $_SERVER['HTTP_HOST'] );

/* That's all, stop editing! Happy blogging. */

It was Varnish.

Our web host had Varnish set up in a Drupal-specific manner, and it was filtering out the cookies that WordPress uses to handle logins.