In Drupal 7, node--[node-id].tpl.php is used as template file for nodes, as well as node--[node-type].tpl.php. Nothing else needs to be done, except, as Clive wrote in his comment, you also need to have node.tpl.php in the directory containing the template files used by your theme.
Notice there is a difference between node--[node-id].tpl.php, and page--node--[node-id].tpl: The latter is also used for pages at node/[node-id]/edit, or node/[node-nid]/delete. See the comment at the beginning of theme_get_suggestions().
Build a list of suggested theme hooks or body classes in order of specificity. One suggestion is made for every element of the current path, though numeric elements are not carried to subsequent suggestions. For example, for
$base = 'page', http://www.example.com/node/1/edit would result in the following suggestions and body classes: