I have a lot of antiques and collectibles and I've always packed the china and small things and moved them myself. I hired North American to move the big furniture and they were outstanding. They moved all my antique furniture without a scratch. They were good to work with and worked very hard. I've used them twice for short moves and would do it again in a heartbeat.
I've never heard of any problems with large moving companies, but I have heard of a lot of people doing online research instead of talking to anyone and getting taken by a "company" that isn't licensed or even really a moving company. I stick with the large moving companies and check credentials when they arrive. I've always had good luck with my 3 moves in the last 20 years. Five if you count helping my mother move into and out of assisted living.
Also, be sure whoever you use it insured and bonded. If they are, you have recourse should anything go wrong. Insured companies will cost you more, but they're worth it.