varnish + multiple webheads = files/(css/*.css|js/*.js) missing
This is happening when you have optimization for CSS/JS turned on. These files are generated regarding the context (the page visited by the browser) and so the visitor won't be served some files when the round robin changed the targeted server.

You can configure the round robin to stay on the same server on a per-session basis (see: Pure VCL cookie-based sticky sessions in Varnish 2.1), but generally you should set up a NFS share between your server where you'll put the sites/default/files folder.

Source: Drupal's optimize CSS and JavaScript setting causing problems in multiple server environments.