Downsides To Opening & Closing Too Many Credit Cards

Credit card sign-up bonuses have been improving this year, but there's a concern that applying for too many credit cards have downsides like a hit to your credit score. My Money Blog has a good review of this and shows that the credit score impact is minimal. He's actually more concerned with another aspect of applying for too many credit cards:

Based on my own experiences, my personal choice is to limit myself to about 3-5 credit cards per rolling 6-month period (more when there are good offers and less when there isn’t). My primary concern is not really the credit score, but more the fact that individual issuers might not approve me just due to the inquiries even though my score is fine.