This doesn’t mean your case will go away. All this means, is that any incriminating statements that you had made cannot be used against you in court. If the officer wants to ask you questions other than your name and address, s/he will then read your rights. At this time, you do not have to answer any questions because they will be used against you in the court of law.
A tactic that law enforcement sometimes use is asking you questions before reading you rights, getting you to admit something and then reading you your rights and asking the questions again. For the most part, the person under question would assume since they already made a confession or incriminating statement, the officers already know the truth so how else could it hurt them? This is false. Since they got you to admit something after reading you your rights, all statements thereafter will be used against you. Anything you say before your rights are read cannot and will not be used as incriminating evidence. The most important thing you can do while being investigated after an arrest is to call a criminal defense lawyer and make no statements before doing so.