How can I allow a user to add hyperlinks into a JTextPane? Such hyperlinks should behave like those found in JEditorPane with HTML documents.

Well you may allow the user to enter the text and the URL in a dialog. Then implement a method -

private int linkID = 0;
public void addHyperlink(URL url, String text) {
  try {
    SimpleAttributeSet attrs = new SimpleAttributeSet();
    StyleConstants.setUnderline(attrs, true);
    StyleConstants.setForeground(attrs, unvisitedColor);
    attrs.addAttribute(ID, new Integer(++linkID));
    attrs.addAttribute(HTML.Attribute.HREF, url.toString());
    doc.insertString(doc.getLength(), text, attrs);
  }
  catch (BadLocationException e) {
    e.printStackTrace(System.err);
  }        
}

public class LinkController extends MouseAdapter 
                            implements MouseMotionListener {

  public void mouseMoved(MouseEvent ev) {
    JTextPane editor = (JTextPane) ev.getSource();            
    if (! editor.isEditable()) {

      Point pt = new Point(ev.getX(), ev.getY());
      int pos = editor.viewToModel(pt);
      if (pos >= 0) {                    
        Document doc = editor.getDocument();
        if (doc instanceof DefaultStyledDocument) {
          DefaultStyledDocument hdoc = 
            (DefaultStyledDocument) doc;
          Element e = hdoc.getCharacterElement(pos);
          AttributeSet a = e.getAttributes();
          String href = (String) a.getAttribute(HTML.Attribute.HREF);
          if (href != null) {
            if (isShowToolTip()){
              setToolTipText(href);
            }
            if (isShowCursorFeedback()) {    
              if (getCursor() != handCursor) {
                setCursor(handCursor);
              }
            }
          } 
          else {
            if (isShowToolTip()){
              setToolTipText(null);
            }
            if (isShowCursorFeedback()) {    
              if (getCursor() != defaultCursor) {
                setCursor(defaultCursor);
              }
            }
          }
        }
      } 
      else {
        setToolTipText(null);
      }          
    }

  /**
   * Called for a mouse click event.
   * If the component is read-only (ie a browser) then 
   * the clicked event is used to drive an attempt to
   * follow the reference specified by a link.
   *
   * @param e the mouse event
   * @see MouseListener#mouseClicked
   */
  public void mouseClicked(MouseEvent e) {
    JTextPane editor = (JTextPane) e.getSource();
            
    if (! editor.isEditable()) {
      Point pt = new Point(e.getX(), e.getY());
      int pos = editor.viewToModel(pt);
      if (pos >= 0) {
        // get the element at the pos
        // check if the elemnt has the HREF
        // attribute defined
        // if so notify the HyperLinkListeners
      }
    }
  }
}