Over the long course of Roman history, there were surely many times when the Roman government failed to meet the needs of its people. If you are asking about one particular time period, please let us know so that we can give you an answer that refers to that time period.
One of the best-known times when the government failed to meet the needs of the people was at the time just before the fall of the Republic. During this time, the tremendous expansion of Roman territory had led to a huge gap between rich and poor and a situation in which many people felt that their rights were being abused by those who held political power.
For example, during this time, the military was becoming much more powerful. Many military leaders demanded land that they could give out as rewards to their troops. Because the military had power and the common people did not, land was often taken from the common people and given to the military. This led to a great deal of strife within Rome as poor people who had been dispossesed moved into the city. This strife eventually helped lead to the fall of the Roman Republic.
This was one very important way in which the Roman government failed to meet the needs of its people at a very important time in its history.