Sure but you don't have any obligation to keep the cards and pay fees year after year. When the annual fee is coming up on a card I call up and cancel it. Most cards give you the first year free anyways. Plus the no annual fee cards add up if you've been "collecting" them for a while and there's not usually a reason to cancel no annual fee cards.
Total it would be like $300, but all of them are "first year free".
Plus usually you can call and get the second year free too.
Regardless, I can cancel w.e. and the increase in my credit score is worth it. At the moment if I want to buy a house (for example) I get the best rates on loans
Many cards are first year free but quite a few aren't. AMEX stopped first year free on a lot of their cards. Regardless if I'm getting a bonus I'm subtracting any fees from that bonus to see if it makes sense. If a card is $100 annual fee and the worth of the first year is $500 it's a no brainier to get $400.