Avocados can definitely go bad, especially if you do not store them properly. It will be important to make sure that you refrigerate the avocados that you bring home from the store, especially if you are not going to eat them right away. Even in the refrigerator though, avocados will only last a couple of days before they start to go bad. Since you will want to avoid any adverse effects from eating spoiled avocado, it will be necessary to store it properly.
You can also put the avocado that you get from the supermarket in the freezer, which is especially a good idea if you do not plan on eating it for a while. Some people decide to use avocado in certain recipes but need a place to store it until they need to use it. If you are going to be putting this fruit in the freezer, make sure that you wrap it up in saran wrap and put it in a tightly sealed container.