We are continuing to look into the issues people are experiencing with The Pitt for Xbox 360. As of now it looks like the file for the English version on Xbox Live was somehow corrupted. We are currently working with Microsoft to remove the existing file off of Live to prevent any further downloads. Our plan is to replace it with a new file as soon as possible. We hope to have another update for you shortly.
Update: The Pitt DLC is now uploading and should be available for download sometime this evening. When it is up, we'll let everyone know.
If you had already downloaded the "broken" version don't forget to delete it first.
Head over to Major Nelson's post for more information.