Julius Caesar is built around individuals in high positions who are essentially admirable people, but had certain character flaws that lead to their descent. Caesar, Brutus, and Cassius fall during the course of the play. Each of these characters had a certain flaw or flaws that led to their demise. By these characteristics, Caesar and Brutus would be considered to be tragic characters. Cassius would not be considered as a tragic character, though.
Julius Caesar was described as ambitious by Brutus during his funeral speech. He says, "...as he was ambitious, I slew him" (III. ii. 25-26). Brutus is saying how Caesar strongly desired power and that was why the conspirators murdered him. For his ambition; that single reason. In addition, he adds, "...Had you rather Caesar were living, and die all slaves, than that Caesar were dead, and live all free men" (III. ii. 22-23). If Caesar would have received the crown, Rome would have became a monarchy.
Brutus believed that this was not what the citizens of Rome desired. From that, Brutus said that he loved Caesar, but he loved Rome more. That is why he participated in the conspiracy.
Brutus' downfall was caused by his poor decision making, and being gullible. His first mistake was when he decided to join the conspiracy. Second, when the conspirators were discussing their scheme, he convinced everyone to let Mark Antony live. Brutus chose to let him live because he did not consider him a threat at the time, and killing him might give them a reputation of being bloodthirsty. Allowing Antony to speak at the funeral was Brutus' third mistake. Antony then caused this mob of people to riot and essentially cause Cassius and him to flee Rome. His decision to advance to Philippi and battle Antony's forces was his last mistake, which cost him his life. Brutus also was very gullible. He allowed Cassius to sway him to believe what he did, and by doing that, Cassius got him to join the conspiracy to murder Caesar.
Although, Cassius' jealousy, greed, and his lack of care for others led him to his downfall, he was not a tragic character. He was not admirable or but he did hold beneficial qualities that got him what he wanted. At the beginning, Cassius was jealous of Caesar because he was going to be crowned and Rome would be a monarchy. He believed someone else was more deserving of the position that Caesar was going to be given. So, in spite of Caesar, Cassius turned for help to his brother-in-law, Brutus, who was very close to Caesar. By grasping Brutus, he would then be able to carry out his scheme to murder Caesar. Cassius also showed no concern for what others thought. He just desired to do what he wanted. This is displayed when he is discussing his thoughts of Caesar with Brutus. Although, Cassius proclaimed that the conspirators should have killed Antony, and he also believed not to advance to Philippi.
Those decisions may have altered their outcome in the end.
Caesar and Brutus were tragic characters because they possessed very admirable qualities, but also they beared one or two major flaws as well. Cassius, on the other hand, was not a tragic character because he was not admirable and possessed negative qualities.
But, in a sense, he did carry a few major flaws over his others.