Yes you certainly should! Many of my tenants learn that after it is too late.
Click the link to read a great article, "Why Get Renters Insurance? We will Tell You".http://www.mortgagenewsdaily.com/4142006_Renters_Insurance.asp
"While some renters may assume that they are covered by some mythical insurance umbrella held by the landlord, others simply do not associate the need for insurance with their own circumstances until it is too late. They aren't homeowners so they don't need homeowners insurance, right? They do not think beyond that question and many probably don't even know that tenants' insurance exists.
If you rent, look around. You probably have a pretty substantial amount invested within those walls you do not own. A stereo, high-def TV, a business and a leisure wardrobe, all of those small appliances in the kitchen, the electronics in the study; even if your furniture was bought second hand or from WalMart it all adds up and should disaster strike, everything will have to be replaced out of pocket.
And there is the matter of liability. Should your television repairman trip on the front walk he will probably sue your landlord. But if the repairman takes a swan dive over a poorly positioned coffee table inside your apartment or if sweet little Fluffy decides that his ankle looks like lunch it will not be your landlord's problem. It will be yours.
Tenant's insurance covers personal property within a home or apartment against the same types of loss covered by homeowners insurance - fire, theft, vandalism, and water damage (but not flood damage and probably not earthquake damage either) as well as protecting your interests should someone have an accident within your dwelling unit for which you might be held liable."
click the link for the whole article by Glenn Setzer