From what I hear from a family member Dead Man's Chest (in theatres July 7, USA) and Pirates of the Caribbean 3 (2007) were filmed around the same time but I see that it's still in production as of April 2006 (on hiatus until late of summer).

Plot summary for Dead Man's Chest:

Once again thrown into the world of the supernatural, Captain Jack Sparrow finds out that he owes a blood debt to the legendary Davey Jones, Captain of the ghostly Flying Dutchman. With time running out, Jack must find a way out of his debt or else be doomed to eternal damnation and servitude in the afterlife. And as if this weren't enough, the Captain's problems manage to wreck the wedding plans of a certain Will Turner and Elizabeth Swann, who are forced to join Jack on yet another misadventure.

Here's a plot summary for the third movie:

The age of piracy is coming to a close, but not before one final dispute is settled. Will Turner sails to the Far East to find magical charts which will help him to find Davy Jones's locker and save his friend Jack Sparrow. Jack is rescued but... Davy Jones (Bill Nighy), lord Cutler Beckett (Tom Hollander), demonic Asian pirate Sao Feng (Chow Yun-Fat), and commodore James Norrington (Jack Davenport) have formed a wicked alliance that threatens the freedom of Jack Sparrow, Will Turner, Elizabeth Swann, their new ally captain Barbossa (Geoffrey Rush) and all pirates in the world. Emerging victorious requires that Jack lead his crew off the edge of the map and into the uncharted territory on the seven seas. To the Shipwreck island on the end of the world.

UPDATE: Pirates of the Caribbean: At World's End (2007) arrives next May.