Imagine walking into every bank you see and getting rejected for every single bank account you apply for. This sounds fictional, but it is real, I assure you. In my experience working for a banker, I estimate a significant percentage of the people who tried to open a bank account were rejected.
Because these people cashed one single bad check. You see, banks have s system set up to try and protect themselves against people who usually mismanage money or commit fraud. Cashing bad checks, defaulting on loans, etc. This works good in theory, but what happens is that innocent people end up getting caught in the "system." A lot of people who have not committed bank fraud end up being banned from opening a bank account.
Checking accounts for people with a history of banking bad credit are sometimes given out by banks, but usually, it's not the case. You can be utterly assured that if you have a history of financial misdeeds, banks keep a careful record of your past transactions. You may think that you can leave all your financial problems behind with one bank and start fresh with another banking institution, but this isn't the case. Your bad banking history will follow you around.
Some banks might still give you a checking account if you have bad credit history, but it comes down to how bad your "credit" is – ie how bad your Chex System score is. Banks don't look at your credit history score, but rather your banking credit – your previous history of banking. If you have had a few financial mishaps, you can probably still open up a new bank account. But if you have had some major problems like home foreclosure, loan defaults, or have had your account closed by your bank, no bank that checks your previous banking history is going to give you a new bank account.
If you find yourself in this situation then you are going to have to either find a bank that does not look at your credit history, or looking around for a second chance account online. A second chance account is a specific bank account created just for people who are unable to open any other account because they have a history of bad banking credit.
My recommendation for people who can't get a bank account is to try and appeal your Chex Systems entry (you can contact your local bank to find out how) or to get a second chance account. Whatever you do, you want to make sure you get a bank account again as soon as possible.