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There's a lot of variation depending on the details, but researching your mover carefully can pay off. Consumers made 1.4 million inquiries about movers through the Better Business Bureau in 2013, a 17 percent increase from the previous year. Over the same period, complaints about movers fell 7 percent, to 7,306.
The American Moving & Storage Association recommends getting at least three in-person estimates, so you know what's a fair price. (Why in person? So the mover can see just how much stuff you have, and gauge challenges like how easy it'll be to maneuver a big dresser down a narrow flight of stairs.)
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But don't necessarily go for the lowest bid. Check licensing, reviews and complaints with AMSA, the Better Business Bureau and sites like Yelp or Angie's List. That helps limit the chance you'll end up with a problem mover who will show up late, handle your items roughly, or worse—hold your possessions hostage until you agree to pay hundreds or thousands extra.
Once you have an estimate, be sure to ask what, if any, factors could result in a bigger bill. Some, like time spent disassembling furniture, could be solved on your end with a little elbow grease. Others, like unexpected summer traffic en route to your new place, would be important to budget for.
—By CNBC's Kelli B. Grant