It is a primary focus of a democratic government.  By enforcing peoples' rights, they are enforcing the law, the Constitution and the principles enshrined within them.  What more important purpose could government have?  I don't think there are necessarily too many rights.  I do think there are probably too many entitlements.

The role of government in ensuring rights: Government (the people who govern) exists off of taxes.  In order to get those taxes, government provides protection to the taxpayers so that they can have a stable enough society to produce taxable wealth.  People want protection so that they can pursue happiness.  Happiness is adequate food, clothing and shelter for self and family, and knowing that no one is likely to come along and take it away from you.  People pay taxes to government in order to get protection from government.  As long as the taxes are not too high, or the protection too little, a stable, happy society exists.

Government should protect everyone in his freedom to do whatever does not hurt anyone else.

There is no right beyond protection that government can provide to one individual or group without first taking it from another individual or group.  So I would answer the second part of the question by saying, there definitely can be too many rights.

I think much of the answer given to this question will be dependent on which type of thinker is asking it.  History has taught us that bad things are usually not far off when government does not take an active role in ensuring and protecting rights for individuals.  When government goes silent in this capacity, some of the worst crimes are perpetrated.  You can find much in historical record of what happens when individual rights are not acknowledged by a social or political order and what happens as a result.  With this in mind, I don't think that there can be too many rights.  The more individuals who are integrated into the political spectrum and social discourse of expression, the better off a social order is.  Believing in this rights- based paradigm, I am not sure there can be "too many" rights for any silencing of individuals becomes a moral imperative to avoid, in my mind.  Yet, I do think that others could articulate a position opposite of this.

The government is, in my opinion, the only entity that can ensure the rights of citizens.  If, for example, the people in my neighborhood want to throw stuff at my house because they don't want people of my ethnicity living near them, who else but the government is going to stop them?  It would certainly not be a good idea to get my ethnic group to come up here and fight them...

I do not think there are too many rights.  However, I think that there are too many people who think only of their rights and not of their responsibilities.  They think only about what they think they should get and not about what they ought to do (or not do) so as to make the community a better place.