This one seems obvious, but a large number of people who rent don't have insurance to cover the contents of their apartments. Some people really believe they are covered under the building owner's insurance policy if there is a fire or if someone breaks into their home. This is not the case.
If you leave the stove burner turned on and the place goes up in flames, you are responsible for all your own belongings. If you are insured - Great! If you are not you may have some problems. Would you have enough money to start over and replace all your belongings? What if your negligence caused damage in other units in your building? Could you pay for all those losses? It's something to think about.