I can tell you a resounding NO based on sagas told to me via people's PMs asking for help.

Having a checking account or even a checking account + a saving account would not make any difference if your credit card activities are deemed "not for intended purposes", or your credit profile is deemed too risky after analysis of your credit card history.

In fact with Chase, the credit card dept is independent to the banking dept. While your branch manager may be willing to go batting for you in the case you have some adverse actions from the credit card dept, the success rate is very low once the Risk Management / Fraud Dept is involved, all bets are off.

One of the help seekers actually first had all her credit card accounts shut down after her bank branch manager went batting for her to get a new card opened. Then some unrelated events happened (she thought those might be the trigger) a couple months later, her bank accounts were shut down as well. She asked the branch manager to intervene for her once again and was told once the Fraud dept is involved, there is nothing he can do / help. Said person was contemplating to file a complaint with some government agency. I have not heard any more news since, so no idea what might be the eventual outcome.

If all your Chase cards are just opened for the initial bonuses and you have opened a boatload of them in a short period of time, you better start using at least one or two of them regularly. Else the beginner's luck could eventually run out when the system eventually flag the pattern of behavior.

The best policy is to have a no fee card with Chase and keep that card active - Freedom is a very good choice as a relationship builder.