It's not actually adding a scrollbar to a table... it's adding a scrollbar to the div which contains the table, therefore the border of the table should not affect the code. You could change the table to a div, or just enter the text straight into that div, and get the same result.
http://www.morad.info/
<div style="overflow:scroll;height:80px;width:100%;overflow:auto">
<div>
Test<br />
Test<br />
Test<br />
Test<br />
Test<br />
Test<br />
</div>
</div>