I don't, but honestly, in "loyalty" cards I never think about the actual issuer. To me, our Aspire is "that Hilton card", not an amex, so I can't even say my # of amexes. But I have few CCs. I don't chase them.
Yes, I figured that their frown on that plus having just cancelled my Delta (since I don't fly it anymore and finished using my last points) to get a UA card (since for SFO, UA is much more useful) were the 2 reasons for being REJECTED.
But here is an excellent example of how brain-dead these companies processes are. I recently bought HGVC timeshares. I then apply for a HILTON card. Wouldn't an intelligent amex want to think about whether the HILTON card is likely to be the new #1 card for somebody who just bought HGVC timesahres? Maybe? Nah. Too much deep thought required.