If you hire an uninsured firm and someone gets injured on the job, you could be held liable and there is no ceiling on what that could cost you. Most contractors will supply you with their own references, but even the shadiest operators can recruit friends and relatives to pretend they are satisfied customers, so don't rely on their references alone. Also, check them out with your community's department of building inspections. Ask an inspector or two for their opinions. Also, ask the prospective remodelers for supplier references, and more than one.
Most contractors deal with multiple suppliers.
They can't stay in business without a source of supply, so it is common practice for someone in trouble to keep paying one supply house while stiffing everyone else. You're asking for big trouble if you deal with someone who can't pay his bills. Number of bidders - You may have heard the old rule of thumb that says you should get three bids, toss out the high and the low and take the one in the middle. That is too oversimplified to take seriously, but it contains some general principles that make sense. There's nothing magical about the number three, except it seems a reasonable number to manage.