A person can be charged with and possibly convicted of both Federal and State crimes for the same set of events, if they involve violations of both Federal and State laws. Double jeopardy does not bar such a prosecution because they are considers two different crimes, and the double jeopardy clause reads:
[N]or shall any person be subject for the same offence to be twice put in jeopardy of life or limb.
This is not considered the "same offense".
This is true whether or not a plea bargain is involved.
The Wikipedia article linked above says:
The government of the United States and of each State therein may each enact their own laws and prosecute crimes pursuant thereto, provided there is no prohibition by the Constitution of the United States or of the state in question. Such is known as the "dual sovereignty" or "separate sovereigns" doctrine
In United States v. Lanza, 260 U.S. 377 (1922) the US Supreme Court wrote:
The defendants insist that two punishments for the same act, one under the National Prohibition Act and the other under a state law, constitute double jeopardy under the Fifth Amendment, and, in support of this position, it is argued that both laws derive their force from the same authority -- the second section of the amendment -- and therefore that, in principle, it is as if both punishments were in prosecutions by the United States in its courts.
It follows that an act denounced as a crime by both national and state sovereignties is an offense against the peace and dignity of both, and may be punished by each. The Fifth Amendment, like all the other guaranties in the first eight amendments, applies only to proceedings by the federal government, Barron v. City of Baltimore, 7 Pet. 243, and the double jeopardy therein forbidden is a second prosecution under authority of the federal government after a first trial for the same offense under the same authority. Here, the same act was an offense against the state of Washington, because a violation of its law, and also an offense against the United States under the National Prohibition Act. The defendants thus committed two different offenses by the same act, and a conviction by a court of Washington of the offense against that state is not a conviction of the different offense against the United States, and so is not double jeopardy
It is interesting to note that as late as 1922 this court seemed to think that none of the bill of rights had been incorporated into the 14th amendment and thus made applicable to the states. But that would not have changed the decision in this case.