mmmm I have just applied for the Explorer card and didn't get approval on the spot. I have Platinum (applied & received card in Mar'16) and Edge (applied & received a few years ago). SMS has arrived asking me to call them to discuss my application. What's my chance of getting bonus points?
If both of your AMEX cards are credit cards, then they will ask you to cancel one of them in order to get Explorer approved. Bonus points is another story.