I have over 30 accounts with Chase. They're one of the best banks to deal with as long as you have $$$ in the bank. Just walk in and tell them what you need and they'll have NO PROBLEM with your requests.
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"There is no peace without war." -Cortex
This is a highly personal question. Common sense is your friend.
In my case, I spread across banks but never have more than 4 per bank. Don't have too many open bank accounts and open stealth accounts than you can realistically handle. Remember that these two go hand in hand and the answer to your question is more about how many accounts you think you can run as opposed to bank accounts as a whole.