My situation was a little different. I had a Simply Cash up to the limit until the zero interest ran out this January. I applied for Platinum and was declined due to something about payments not being enough on American Express accounts; I was only paying the minimum payment each month. Once the promotional interest ran out and I paid it off, I was approved for the Delta and Premier Rewards Gold card.
As long as you are making large payments, you shouldn't have a problem. I reckon any lender would be hesitant to grant more credit to someone when they haven't paid back what they have borrowed yet.