You're renting, so do you really need to shell out more every month for extra insurance? Let's investigate. Should you spring for the coverage or should you not bother? Read on.
Short answer: yes. Get some. Don't worry, rental insurance is cheap. A recent quote based on my personal stats and address (in expensive NYC) returned a price of under $20 bucks a month for $250,000 of coverage with a $250 deductible. Pretty doable. More good news: if you have safety features in your home like a security system or a fire alarm, it will decrease your premium.
Remember, the insurance that your landlord buys to cover your apartment building only applies to the structure itself, it doesn't cover your personal possessions. If, God forbid, there's a fire in your building and your clothes and furniture are lost, rental insurance will kick in to pay you back.
One important distinction: When shopping for rental insurance, you can choose a policy with replacement cost value or actual cash value. The former will cost a little more but will replace your $1000 broken laptop with another laptop; the latter will only give you the cash value of your laptop when it was damaged (usually considerably less then it would take to buy a new one). Choose wisely here.
Another perk: rental insurance can cover more than your things, it can alleviate your liability if someone is hurt while in your apartment or you accidentally cause damage to someone else's property. Policies differ, but most cover a wide variety of risks you face as a renter.
So when should you skip it? You can't be insured by rental insurance if you're living in "temporary" housing ie. a dorm room, a hotel or sometimes, a sublease (check with your policy on time requirements for qualifying for insurance).