Plenty of FTers (including yours truly) have had many, many credit card accounts over the years and are still getting them.
I have way over ten active accounts and way over 100 closed past accounts and am still getting cards when a good offer comes along. Those millions of bonus miles come in real handy. Thanks FTers for all the great info on these bonus offers.
Gotta love the recent BA 100K for card app and $2.5K spend (expires May 5, so, get hopping!)