Hold It!
You CAN be re-charged with the same crime if the case was dismissed by the DA/SA, if more evidence is collected that proves their case. A case can be dismissed by the Prosecution, known as Nolle Prosse (pronounced nole PROSS), or nolle prosequi; or the case can be dismissed by the Judge, with or without prejudice, often done when certain procedures are not carried out within a specific period of time (for example if the prosecution does not file a formal charge within 30 to 45 days of a person's arrest if the person remains in jail).
If the Judge finds that the case is permanently unprosecutable, such as because there is evidence of tainted police procedures, he can dismiss the case WITH prejudice, meaning the case is forever barred from prosecution.
The DA may also nolle prosse a case if a person enters a pre-trial agreement, similar to probation, and fulfills the conditions of such, such as community service hours and/or restitution. If this happens this is part of a plea agreement, and upon satisfactory completion, is forever barred from prosecution.