Yes, banks can refuse to open an account for someone they believe is risk for someone. Banks are businesses, not arms of the government (even if they are regulated and bailed out by the government); as private businesses, they have substantial discretion to not do business that they believe is unprofitable or risky--such as giving an account to someone who previously had defaulted on an overdraft. You should try other banks; you should also try to raise your credit score, such as by taking secure credit cards, using them modestly, and paying them promptly. If you do a few web searches, you should be able to find alot of advice on how to raise your credit scores. Good luck.