In federal court, the sentencing judge is not bound by any sentence agreement made with the prosecutor. The accused is bargaining only for the prosecutor's recommendation as to sentence, So, it was quite legal for the judge to reject the parties' sentence recommendation assuming that when the plea was changed to guilty or no contest, the judge explained in open court to your brother that the court was not bound by the sentence recommendation.
As to whether your brother has grounds for appeal, that is an entirely different question which no one can answer until after reading the entire record of the case. Since your brother had a federal public defender in district court, then he likely is entitled to have the court appoint counsel on appeal. However, you and your family should consider whether you want to hire private counsel for him.
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Grounds for appeal when the accused has pleaded guilty or no contest are extremely limited in federal court. It doesn't generally matter why your brother took the deal (fear of a longer sentence). The time for raising issues of whether the prosecutor had a provable case ("only evidence they had...") was before your brother pleaded and was sentenced.
You may want to discuss this matter with a skilled federal criminal defense attorney in private practice. It is highly unlikely that the Office of the Federal Public Defender will carry an appeal on a matter where the defendant pleaded guilty/no contest and waived his right to trial. You may not have been present at the time of the plea, but you can be certain that before accepting the plea and imposing the sentence, a number of very specific admonitions were given to your brother and he was given ample opportunity to make a different decision. These facts tend to preclude a successful appeal and the Fed Public defender will ordinarily not expend its resources to overturn a conviction by plea.
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