Usually, this is not the server not supporting it, its a workaround that the server provides so that you can consume the API without AJAX from HTML forms, which are restricted to GET and POST.

EDIT: Although in some cases there is an issue that people choose not to configure it on the web server, and instead use headers or other mechanisms to tunnel the "real" method to the application. But most servers do support it, this seems to be a mechanism for routing around administrative issues in organizations.