Symptoms
When several instances of your custom Web part are living on a page, the verbs menu is not displayed when you click on the top right arrow of the Web parts chrome. You can also notice that a JavaScript error is raised when you try to click on this menu.
Cause
The OnInit event has been overridden into your code, but the base method is not called. So, in the JS code produced by WebPartManager, no ID is given to objects controlling the client-side script of your Web parts.
Resolution
In your code, make sure that base.OnInit(e) is present in the OnInit method.
protected override void OnInit(EventArgs e)
{
// Your code …
base.OnInit(e);
}