No, and yes - sort of, in order.
The teaser system assumes that there will always be more to read so if you choose to show the teaser, by default it will always have a 'read more' link.
You can, however turn off the link entirely quite easily. Copy node.tpl.php from your main theme into your custom theme directory and comment out this part:
<?php print render($content['links']); ?>
If you only want to show a read-more link when there is more to read, you almost certainly need to build a custom module for that. There's lots of good documentation on how to do that, but we can probably help with the specifics here once you have a framework going if that's what you decide to do.
Note that if you only want to change the behaviour for a particular content type, put some logic around the line instead of commenting it out, like this:
<?php if ($node->type != 'article'): ?>
<?php print render($content['links']); ?>
<?php endif; ?>