Carl Sagan's Contact is a good story. I think the main part of both the book and the film was highly believable. Everything about how we would handle a situation like this is very believable; the lack of trust between countries, the government trying to cover up the truth in the end, the religious aspect and how women are seen upon in the scientific - and man dominated - world.
The book focused much more on the breaking of the code (in the movie that part was done in about five minutes I think), and it was really interesting to read about how they were trying to solve it. If we would get a message from space for real I think it would be a real challenge to decode it, but I don't think the message would be coded by the sender deliberately to be hard to break. The challenge will instead be for us to recognize the message as a message at all. To separate it from all the other radio noise in space.
What I don't find so believable about the story is the thing about traveling in space through worm holes. I think it is strange - and that he totally ruins the story - when Sagan suggests that it would be possible. Even if it even would be the case that worm holes do exist - I know many scientists have written lots on this - I just can't believe a human or anything organic could survive passing through them. I think it is equal to suggesting we can travel through a brick wall - like it's done at the railwaystation in the Harry Potter books! J.K Rowling may get away with it but not Carl Sagan!
The book and the film's concept up to that point when the journey starts is really believable but it would not have been necessary to go past the point when they (just Ellie in the movie) go away in the machine. I think it would have been better if both the book and the movie had ended right there and left us wondering. Just like the 70's movie Close encounters of the third kind did.