It seems that renter's insurance only covers the tenant's property and perhaps guests. I suspect that a far bigger financial risk for a tenant is in liability due to causing a fire, flood or other damages. How can one get insured for that?
As specific examples, your desktop computer could spark and catch fire (this does happen), or the drain pipe inside the wall may decay and break, flooding several floors below (this happened to me), and you could be accused of using the wrong drain cleaner, even if it is some of the previous residents' doing.
The regular renter's insurance policies that I've seen do not claim to protect you in these cases, so I have to assume that they don't.