Potatoes do eventually go bad. Their exact shelf life depends on the type of potato, but most go bad after two to five weeks at room temperature or two to four months in a refrigerator.
Over time, potatoes eventually soften and wither. They can also grow mold if they are stored wet. If there is sufficient humidity, a potato eventually grows sprouts. These sprouts are highly toxic, as are green patches that sometimes appear when the potato is exposed to light. The rest of the potato can be eaten safely once the sprouts and green patches have been cut out, provided the potato is not moldy.