I have cards with all the major banks:

AMEX: 3 personal, 1 business
Citi: 4 personal, 1 business
Chase: 6 personal, 1 business
Cap1: 1 personal
Barclays: 1 personal
BOA: 2 personal
HSBC: 1 personal

This does not count 6 - 8 department store cards. I manage them using Quicken and pay them off monthly.

I have several cards with an annual fee that I continue to renew simply because the perks are worth more that the fee. The others I cancel after the annual fee posts and wait 2 years to apply again. I generally keep a few of the no annual fee cards and make a few charges each year to keep the card open and improve my credit score.

Most banks don't limit you to a certain number of cards, instead they limit your available credit. Most of the time, especially with Chase, if your app is denied you can call the reconsideration line and move some of your credit line from another card to the card you want to open.