<p>I have a JComponent that's painting various shapes on itself.  I'm detecting whenever the mouse enters one of these shapes and changing the tooltip accordingly.</p>

<p>The problems I'm having are:</p>

<ul>
<li>The tooltip doesn't follow the mouse as the user tracks the mouse across the shape. It 
stays where it was first set and then only jumps whenever another shape changes the tooltip.</li>
<li>It takes about a second for the tooltip to appear, but I'd like it to appear immediately.</li>
</ul>

<p>Can someone suggest a way of getting these behaviours without writing a custom tooltip mechanism?</p>

Take a look at the
ToolTipManager
.
You can register your component with that manager and then adjust a number of settings. Its pretty straight forward to use.
That at least can solve your initialdelay problem.
For your first problem you can overide the createTooltip command from your component to get a hold of the JTooltip instance. and then its easy make the position change whenever you move your mouse(aka follow your mouse) as its a subclass of the JComponent class.