I don't know about other banks, but I do know Chase will definitely consider your banking relationship when it comes to CC apps and CLI requests. Obviously, the higher your balances, the better. Generally, you need at least $10k on deposit for it to really matter. Once you get above $5mm, you can start to do really creative things on the credit card front (e.g. getting approved for a six figure credit card limit when you have a 600 FICO).
However, as someone else said, the possibility of CLIs should be very low on your list of criteria to choose a bank.