EDIT:
Shuffle the bound values set the bounds only of ScrollPane
and not of TextArea
.
JTextArea _area = new JTextArea();
// _area.setBounds(10, 10, 400, 250);
add(_area);
JScrollPane scroller = new JScrollPane(_area);
scroller.setBounds(10, 10, 400, 250);
scroller.setVerticalScrollBarPolicy(ScrollPaneConstants.VERTICAL_SCROLLBAR_ALWAYS);
add(scroller);