Run up bills is a phrase used to describe the act of accumulating debt. It is usually used in reference to someone who has been spending more money than they can afford, resulting in them owing money to creditors. The crossword solutions for this term are OWE (3 letters) and SPEND (5 letters). Owe is a verb that means to be indebted to someone else, while SPEND is a verb that means to use money to purchase goods or services.