Two factors will generally determine whether you can get your case dismissed for the failure of the police to give you Miranda warnings: 1) how your statement was taken, and 2) the other evidence against you. Miranda warnings refer to a set of warnings which advise you that you have the right to remain silent and to have an attorney during questioning, and that any statements you make can be used against you in court. Most people think that the police have to read your Miranda warnings every time they talk to a defendant. However, the police are only required to read you Miranda warnings before custodial interrogations. Custodial interrogation means that you are in jail or in custody and the police are asking you questions about a crime. If you are not under arrest or in custody, the police are not required to read you any Miranda warnings.
If you were in custody or jail and the police asked you questions without telling you about your rights and refreshing your Miranda warnings, any statements you made were obtained illegally. If you or your attorney files a motion to suppress, the illegally obtained statements should be thrown out. Keep in mind, though, that only the statement gets thrown out. Your case may or may not be dismissed depending on the other evidence available. For example, if you confessed to assaulting your girlfriend and the statement was thrown out, the state may still continue with the prosecution if they have other evidence, like an eye-witness. If the only evidence against you is your confession, and it is thrown out because the police didn't give you Miranda warnings, then your case will have to be dismissed.
Miranda warnings stem from Supreme Court rulings. Either through statutes or interpretations by courts, some states actually provide you more protections and rights. The best defense to illegal questioning, though, is silence and a good attorney. Many people will attend interviews and answer questions in a vain attempt to cooperate, thinking they are helping their case. Unfortunately, they end up creating more headaches for themselves and their attorneys who try to defend them later on. If you think that you were tricked into giving a confession, you should consult with an attorney as soon as possible to see what rights are available to you in your jurisdiction.