Do your customers trust your apology?

In this NYT article, author Daniel Pink challenges businesses to speak like real people.  The whole article is worth your time, but what grabbed me was this simple quote from the head of the t-shirt site Threadless:

The best way to figure out if you're running a good company is to figure out if your customers trust your apology.

I think this is right on, and a great measure for every business.  Do your customers trust you when you apologize to them for making a mistake?  You do apologize, don't you?