Lot's of people break them, and only a very small minority ever end having to pay anything.
It depends on the contract, the conditions under which it was signed, and which state has jurisdiction.
If you owe a small amount, you could just take a chance on having to pay it. If your in for $10K, you might want to talk to a lawyer first...he could probably tell you how to dodge paying if the contract is less-than-airtight (likely).
Note: Weigh the dollar amount vs. having a PO'ed previous employer as a reference. If it is a disreputable company (mesa), nobody probably cares what they think. A small company might take it personally, and be out to get you down the road.