Generally you can only insure up to $25,000 for personal property, and claims are valid only for major damages (fire, accident, etc.).  There are definite limits to coverage.  For example, if your grandmother's cut-glass bowl gets broken because you have to slam on the brakes, that's a packing problem not covered by most companies. 

The day you reserve a moving truck is a good time to review your homeowner's insurance policy with your Insurance Agent to determine if it covers your possessions during a move. If it doesn't, you can ask if there is a separate rider you can add to your policy for the transit, or you can usually buy cargo insurance through the truck-rental firm. Cargo insurance protects against accidental loss or damage to your possessions while in the rental truck. 

Whichever method you use to move, be sure to save receipts. If you are moving in conjunction with a job, and itemize on your tax returns, you can probably write off moving expenses at tax time. Consult your tax accountant to be sure.