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I once spoke with a rep saying you don't necessarily have to pay by this date, but instead by date X (before they finalize the statement). Is there any disadvantage to submitting a payment for your balance after the Please Pay By date?
Other than an increase risk of missing a payment, there aren't any.
Your "Please Pay By" date is a suggested payment date that's 15 days after your statement cuts. You have up to the day before you billing cycle ends to pay. This, of course, is if you do not have any "Pay Over Time" charges (should you be eligible and are using).
Is there any downside to how AMEX would view me? Been a customer with them for about one year and everything has been great. Not a big spender by any means, but I would say my relationship with them is good.
Nope. Unless you're late or miss your payment. In theory, your spending (and payment) habits will dictate how much spending power is alotted to you on the card. Use it more, and pay it often, you won't have any issues. AMEX is a great creditor as long as you're a great customer.