Jonathan I had a similar issue trying to remove a card. I suspect you like me only have the one card associated with your paypal account. When that was the case for me it did not allow the option to remove the card information.
As my goal was simply to change what card was associated with my account, I had to first add the other card as a backup. Once there were two payment methods associated with the account it gave me the option to delete a payment method, and I then deleted my old card.