A secure phone line is conceptually possible; this is not really different from, e.g., a secure communication between a Web browser and a HTTPS server (there are technical subtleties about lost packets and whether they should be tolerated, but that is not the issue here). However, the movie-secure phone is not secure, and that's a structural problem.
The problem lies in the question: "is your connection secure ?". If you need to ask to the other guy whether the line is secure, then the line is not secure. That's as simple as that. A "bad guy" could hijack the line and, when you ask whether the connection is secure, the bad guy could simply respond "yes it is !", counterfeiting the voice of the intended recipient (and, in the other direction, he could "replace" the question by an innocent sentence).
In a secure phone line, the caller and the receiver shall be authenticated to each other, which can be done with various cryptographic tools (e.g. digital signatures, or, more simply, a shared secret). Variants of the same tools also establish a session-specific shared secret which can be used to symmetrically encrypt the data. Bottom-line is that once people begin to actually talk, the line should already be secure and both participants shall have ways to know it (e.g. they are using special phones which refuse to communicate if security is not achieved). Otherwise, there is no security.
On a more practical point of view, if I were to implement a phone-like secure system between two entities, I would investigate using VoIP over a VPN. This would require some delving into the details of the VoIP protocol, so I would do that with an open protocol (i.e. Ekiga, not Skype).