The Civil Rights Movement was very successful in moving America towards legal racial equality, but not very successful in moving the country towards actual racial equality.
The major legal goal of the movement was to end legalized segregation in the United States. This goal was accomplished with the passage of the Civil Rights Act of 1964. Since then, there has not been any legal segregation and overt discrimination has faded as well.
However, the Civil Rights Movement did not do very well in terms of making blacks and whites actually equal. It did not succeed in giving blacks the same education opportunities as whites have, for example. This continues to show up in the fact that blacks are much less wealthy, on average, than whites.
The Civil Rights Movement was able to end segregation and most discrimination. However, it was not able to make up for the impacts of past injustices and it has not been successful in creating a truly equal society so far.
My parents attended segregated high schools their freshman year. I often tell my students that they are the first generation to grow up in an America where the idea of segregation is completely absurd. So the movement itself was critical in moving us in this direction, but as all of our history since the Civil War has taught us, moving a country forward socially is a generational process, and a continual one. It is slower, more complex and more difficult than any one movement or organization can change on their own.
The same was true of the women's rights movement. It was my generation that began accepting women into the mainstream of the American workforce, and today, the idea of a female President is perfectly acceptable to almost everyone. Perfectly inevitable in our society. These are changes that occurred over long periods of time. The Civil Rights Movement of the 1950s and 60s was critical in moving us forward, but society responds to such advances and changes in its own time.