The major success of the Civil Rights Movement was the destruction of legalized segregation and discrimination. This can be seen in the three greatest achievements of the movement. There was the Brown decision which made school segregation illegal. There was the Civil Rights Act of 1964 that made discrimination in public accomodations (both in terms of service and employment) illegal. Finally, there was the 1965 Voting Rights Act that protected the right of African Americans to vote.
However, the movement had a major failure because it did not eradicate inequality of opportunities. Even once the laws were changed, African Americans were still disadvantaged. They were much more likely than whites to be poor and to live in poor neighborhoods. This made it hard for them to get good educations for their children and ended up ensuring that inequality would continue even after legal equality was gained.
It might be asking too much of the movement to say that it failed because it did not eradicate inequality. However, that inequality is the biggest problem facing African Americans today because it did not go away when legal equality was achieved. In that sense, the Civil Rights Movement failed to gain true equality for African Americans.