What mistakes are made regarding Brutus—either that he makes himself or others around him make, in Shakespeare's Julius Caesar?
Perhaps the most destructive of the mistakes made in Shakespeare's Julius Caesar is when Brutus joins Cassius and his group in their plan to assassinate Caesar.
Brutus is a man who loves Rome: he would do anything to protect the empire. In his speech to the Roman people, Brutus makes this clear...
...Not that I loved Caesar less, but that I loved Rome
Brutus mistakenly trusts Cassius, even though he questions Cassius' motives when Cassius first approaches him:
Into what dangers would you lead me, Cassius... (I.ii.70)
Cassius simply resents the way Caesar has treated him, even though Cassius saved Caesar's life. Cassius is motivated by Caesar's treatment of him: Cassius insists that he and Brutus were born free, and equal to Caesar, but are treated poorly, and he pushes Brutus. Brutus acts for Rome, but allows himself to be pulled into Cassius' plot—which is of no benefit to Rome...for Cassius really only cares about himself.
In their plans, Cassius and the others tell Brutus that it would be wise that Marc Antony die as well as Caesar, but Brutus resists, and this is a mistake. He argues that it is enough to kill Caesar—Antony is simply one who does Caesar's bidding. To also kill Antony would make them "butchers."
Let us be sacrificers, but not butchers, Caius. (II.i.166)
Cassius is correct: Antony is as dangerous to them as Cassius suspected. Even after he has pledged his support of Brutus and the others, Antony promises (when the others are gone) to avenge Caesar's death, noting that the spirit of the murdered Caesar will...
Cry "Havoc!" and let slip the dogs of war… (III.1.273)
It will be Antony that will throw the entire country into civil war.
Brutus is an honorable man: dedicated to his country, valuing it above his own life. But he is idealistic—believing that the murder of Caesar will solve the ills of Rome. It is not possible that the murder of Caesar, a violent and traitorous act against a man Brutus loves, could end in anything of a positive nature for Brutus or Rome, and Brutus is mistaken to think good could come from this act, or that killing Caesar could leave him unscathed.