To add rights is to subtract rights; to not allow punishing evil is to punish good.
For example, let's say I believe in self-defense, but have a quirk in my concept of what counts as aggression: it can't be aggression if it doesn't involve physical contact. Thus, if Alice steals Bob's phone while he isn't holding it, I don't see Alice as acting in aggression, but the disturbing consequence is that if Bob retaliates by beating her to get it back, I don't see him as acting in self-defense; I see him as acting in aggression because he's using violence in response to something I didn't see as violence. (This isn't even a made-up example - though usually not made explicit, this specific misdefinition of aggression is common in stories and in dealing with children.)
This is why it's so very important to understand what is aggression and what isn't. If you don't see something as aggression that you should, you will at some point see self-defense as aggression. You can't have one error without the other.
Another example: if I believe that "health care is a human right", what does that mean for me when people don't have health care and can't get it by themselves? It means that someone else has to provide it for them or pay for it to be provided, and they don't have a choice, even if they're not responsible for the sick person's condition. In the mildest case, I'll steal from someone else to get health care for the person who doesn't have it. Thus, by adding the right to health care, I necessarily subtracted or demoted the right to keep the fruits of your labor.
The modern American (statist) mind doesn't seem to realize this. People will say that others have the right to keep their property, and also have the right to affordable education, health care, housing, and an array of other things, and not notice that these can't all be upheld at the same time. The only meaning of saying someone has a right to X is that if they don't have it, someone else must provide it. For these other rights to exist, private property has to be demoted.
Now as a non-libertarian-purist I have to admit that I believe private property is negotiable when conflicted with other moral concerns, and that it can be justified to violate property rights in a dire situation (although I still wouldn't explain this as people having a right to life because I use 'rights' to refer to only part of a complete moral system). But we have to be conscious of this, and not think that adding rights actually changes the amount of rights we think people have.