Extra insurance, photos before and after, opening and checking before signing off--all excellent suggestions and I would follow each one. After making more moves than I can count on both hand (and feet!!!) there have been great experiences, okay experiences and a couple of 'Oh Crap' experiences.
I think it comes down to how vigilant you are during the pack-up. It really depends on WHO is doing the packing. There is a very different level of responsibility if you are doing your own packing or if the moving company is packing for you.