It's going to totally depend on the restaurant.

I know one local place goes nuts when they have a certain type of fish coming in, usually sending out e-mails a day in advance saying they will have it the next day (not a fish I like, so I don't pay much attention)

If a restaurant has shrimp on the menu, and if they have calamari on the menu, if they were to offer a shrimp and calamari special over pasta, I'm not going to think they are just tryng to use up old food, as it's regular menu food. It's just a different way of trying something.

I've also seen more than a few chefs hitting a farmers market in the morning (Union Square, New York for example) and had the chance to chat with them sometimes as to what they were planning on what they were buying, and in many cases it was for the the daily specials.