Thats not an SSCCE!

You have a bug in the code you are *not* showing us.

Here's a working SSCCE based on your snippet:
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import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.KeyEvent;

import javax.swing.JFrame;
import javax.swing.JLabel;
import javax.swing.JMenu;
import javax.swing.JMenuBar;
import javax.swing.JMenuItem;
import javax.swing.JPanel;
import javax.swing.KeyStroke;
import javax.swing.SwingUtilities;

public class TestAccellerator implements ActionListener {

    TestAccellerator() {

    // So I don't need to edit Chameleon's code at all
        ActionListener listener = this; // ick.

    // Chameleon's snippet *exactly* as in newsgroup
        JMenuBar jmb = new JMenuBar();
        JMenu jm = new JMenu("Help");
        JMenuItem jmi = new JMenuItem("Help Index");
        jmi.setAccelerator(KeyStroke.getKeyStroke(KeyEvent.VK_F1, 0));
        jmi.addActionListener(listener);
        jm.add(jmi);
        jmb.add(jm);
    // end of Chameleon's snippet

        JPanel p = new JPanel();
        p.add(new JLabel("Panel"));

        JFrame f = new JFrame("Test Accellerator");
        f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        f.add(p);
        f.setJMenuBar(jmb);
        f.pack();
        f.setVisible(true);
    }

    public void actionPerformed(ActionEvent e) {
        System.out.printf("Event: command '%s' from object '%s'\n", e
                .getActionCommand(), e.getSource());
    }

    public static void main(String[] args) {
        SwingUtilities.invokeLater(new Runnable() {
            public void run() {
                new TestAccellerator();
            }
        });
    }

}

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When I press F1 I get the output I expected.