If ambition is defined in terms of personal ambition, self-advancement, then there is no compatibility between ambition and honor in the play. Cassius' desire for Caesar's death is one of personal ambition; he despises Caesar and resents Caesar's achieving such great power in Rome. Cassius makes no attempt to hide his bitterness:
. . . and this man [Caesar]
Is now become a god, and Cassius is
A wretched creature, and must bend his body
If Caesar carelessly but nod on him.
Cassius seeks to bring Caesar down for one reason: to gain power himself. He deceives and manipulates Brutus shamelessly to accomplish his ends. Once he has gained power, Cassius uses it to enrich himself, for example, by accepting bribes. There is no honor in Cassius.
Caesar himself is portrayed as a man of personal ambition, as well. After assuming power in Rome, his actions implied that he acted out of ego and self-glorification rather than honor or love of country. It was Caesar's obvious love of power that troubled Brutus deeply and made him vulnerable to Cassius' attempts to draw him into the conspiracy to murder Caesar. Caesar's imperial nature is expressed clearly in Act III:
I [Caesar] could be well moved, if I were as you.
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But I am constant as the Northern Star,
Of whose true-fixed and resting quality
There is no fellow in the firmament.
Caesar does not speak of honor or commitment to Rome; he speaks of himself, always.
Brutus' ambitions, however, are not personal in nature, and he is the only character who acts out of a sense of honor. Early in the play, Brutus identifies honor as the primary value which guides him, affirming that he loves honor more than he fears death itself. Throughout the play, Brutus remains true to his principles; his actions are motivated solely by his love of country. Antony pays tribute to Brutus' honorable character at the play's conclusion:
This was the noblest Roman of them all.
All the conspirators, save only he,
Did that they did in envy of great Caesar.
He only, in a general honest thought
And common good to all, made one of them.
Studying Brutus, Cassius, and Caesar, we can conclude that ambition and honor are not at all compatible in Julius Caesar.