Moving quotes should be made by an experienced agent from a professional moving service. If you use a cheap moving company, they are likely to ask you over the phone how much you want them to move and give you a guesstimate based on that.
Most professional movers will give you a moving quote for free. Someone will come and appraise your situation and belongings, and this agent should be able to figure out just how difficult it will be to move pieces of furniture and how many boxes you'll need. This can even help you decide how many boxes to buy--it's a real annoyance to make multiple trips to get boxes if you're packing at the last minute and find yourself short, but no one wants to buy more than he needs because boxes are expensive.
While you should be aware that a moving quote is not a final cost, that doesn't mean you should be at the mercy of paying professional moversmore if they demand it on the day of your move. Some less reputable movers will do that solely because they know you're likely to give in because you wouldn't know what to do if they left.
When you get your moving quote, you should have it itemized so you know what the costs are attributed to. Then, if the actual cost ended up being more, you can see a correlation between the higher cost and the greater number of boxes or greater weight. If the discrepancy is disproportionate, you can speak with a supervisor or agent from the moving company who should tell the movers to work for the right price