From the facts you have set out, it sounds like the renter's insurance will cover the loss to the limits of the renter's coverage, then the landlord's insurance will cover the rest. At that point, the landlord's insurance company will decide whether it is willing to pursue the renter for the difference.
While it is possible to pursue the renter personally, many times it is not cost effective to the insurance company to do so.
While I'm not a Nebraska attorney, in general you will need to look at the lease agreement. This will tell you whether there is any prohibition preventing landlord or its insurer from suing the tenant. Some leases for example limit the right of recovery to insurance proceeds. Absent any such prohibition, you would generally have a right to sue the tenant for any excess amounts. If he has no assets or income, that means that while you may win the lawsuit, you will have no access to the awarded damages i.e. you can't get blood from a stone. However a judgment against him may be worthwhile in case he ever comes into money or has income.