For keeping a film a secret, it depends. I mean, with Cloverfield, they were trying to make sure that no one knew about the film during its production. But the biggest problem with keeping a film in secret nowadays is the advancement of communication technology. Though you can not make an announcement of a film being in production, if you're shooting a film somewhere, there is always the possibility that someone in a nearby building or an onlooker will manage to snag a picture from the set while in the production. I remember seeing a picture of Angelina Jolie during the production of Salt that was taken by someone up in a building. There were a couple of pictures of a scene that involved Sandrine Holt in Underworld 4 that was posted on Flickr by someone who saw the shooting going on before they got pulled. And I recall footage of what was a part of a scene from the film Super 8 while it was in production as well, shot by someone who lived in the town the movie was being filmed in.
If you could control the location well enough that no one would be able to get a clear shot of a shoot in progress, or misinform the people of what the movie that is being filmed is, then it's slightly possible. As for 100% possible, not so much anymore due to communications being much quicker nowadays.
As for stuff seen in screenings, maybe, if the surprise is actually left out of the preview cut.
Not to mention, some of the rumors and info that we hear about production is often purposefully designed to be leaked to build up interest, while the other half of the time its not. Think about it like this: The recent Dark Knight Rises trailer was leaked. Now only people who work at Warner had access to the trailer. The only other people outside of Warner that had access were the the trailer editing company and Christopher Nolan himself.
So, the real question is: Is it possible for filmmakers and studios to manipulate information to gain interest? The answer is: it's possible.