I've done it, and continued to live in the old (paid for) house for about a year until we got the new one livable. Even took out a home equity line of credit on the old house to pay for some work on the new house.
But both houses cost less than we make in a year so the financing wasn't hard to pull off at all.
My one piece of advice though would be to pay attention to taxes- in MI, your primary residence is taxed at a lower rate than any other residences you own. I should have made the more expensive house our primary residence. Overall, it probably didn't cost me more than a $1k, but that did really screw up our escrow numbers for a couple years. Some localities allow you to have two primary residences if one is on the market- all worth investigating.