Your question assumes that the plebs did, in fact, fail. The plebs (or at least populists, whether they were personally of plebeian background or not) may not have gotten everything they wanted in their attempts to reform the Roman constitution (the tribal assembly was still pretty rigged toward the patrician families), but they certainly changed it a lot from its initial inception to include a lot more participation from the plebs. By the late Republic, in fact, there were some offices reserved for plebs (tribunes of the people), while there were no offices which were actually exclusive to patricians. (The way that the Roman tribes voted in assembly was still rather rigged in favor of the aristocratic gens, though.)

The plebs mostly got what they wanted, effectively, through strikes. In 494 BC, the office of Tribune of the People was created because the plebs refused to march to war. The Hortensian Law was also passed in response to the plebs effectively striking on the Janiculum hill.

On the whole, by the late Republic, conflict between the old blue-blooded aristocracy and the plebs was no longer really the central focus of class conflict. The main conflict was more between the rich and poor -- with the rich including many wealthy plebs. Small farmers were becoming rarer and rarer, and the countryside was becoming dominated by great estates worked by slave labor.

Your next question might be "were there any attempts by the Roman poor to overthrow the rich?" That leads to another line of inquiry. I don't want to go into a tangent on that, but I'll just say that yes, there were (heavily resisted) attempts to reform the distribution of wealth, not just aristocratic privilege, by the late republic, and these were less successful than the attempts of the plebs to achieve political equality with the patricians.