I have many miles/points-related cards and a handful of cards that I picked up specifically to use to make money with 0% teaser rates.

The Banks created this game, so they are certainly happy to have you play. I have a fistful of Citibank, Bank One, MBNA, and Chase cards.

Number of cards plays a small factor in your credit score. I've noticed mine moving slightly down when I have a flurry of new signups for bonuses, 0% teasers, etc. But it's a minor factor: your debt ratios and payment history are much more important factors.

I probably wouldn't play these games at the same time I was shopping for a mortgage, but I don't worry about small movements in my credit score thanks to picking up a new card: it's just not that big of a deal.