Actually people under 18 CAN get a checking account.. and usually banks will only turn you down if your in chexsystems, which basically means you've screwed up with another bank, and usually if you had an overdrawn account that was charged off, or if you have a long history of returned checks.
Some banks offer "second chance accounts" which is basically just that, a second chance to prove yourself worthy of having a bank account, these accounts usually come with some restrictions but once you keep the account in good standings for a certain time period, usually about a year, the bank will choose to "graduate" you to a normal account. Try Wells Fargo for their "Opportunity Account"
Banks have a special database where they store the identities of people who bounced checks in the last 5 years. If someone did that then many banks will just not allow it. Besides that there are some picky banks that want to run credit checks on everyone who applies for a checking account. If you have bad debts on your credit report then they can deny you too. Best bet is to ask the bank what their official policy is.
People who have overdraft accounts that have gone to collection with other banks, people under 18 (without a co-signor)
Under 18 or past bad checks that are recorded on ChexSystems. Apply online if you're worried about getting turned down. Key bank is one.