Index: wp-content/themes/twentytwelve/comments.php
===================================================================
--- wp-content/themes/twentytwelve/comments.php	(revision 22580)
+++ wp-content/themes/twentytwelve/comments.php	(working copy)
@@ -45,11 +45,17 @@
 		</nav>
 		<?php endif; // check for comment navigation ?>
 
-	<?php // If comments are closed and there are comments, let's leave a little note.
-		elseif ( ! comments_open() && '0' != get_comments_number() && post_type_supports( get_post_type(), 'comments' ) ) :
-	?>
-		<p class="nocomments"><?php _e( 'Comments are closed.', 'twentytwelve' ); ?></p>
-	<?php endif; ?>
+		<?php
+		/* If there are no comments and comments are closed, let's leave a little note, shall we?
+		 * But we only want the note on posts and pages that had comments in the first place.
+		 * It's just the polite thing to do!
+		 */
+		if ( ! comments_open() && ! empty( $comments_by_type['comment'] ) ) : ?>
+		<!-- If comments are closed. -->
+		<p class="nocomments"><?php _e( 'Comments are closed.' , 'twentytwelve' ); ?></p>
+		<?php endif; ?>
+
+	<?php endif; // have_comments() ?>
 
 	<?php comment_form(); ?>