Depends on your bank. I have credit cards with 2 different banks. At first both of them flat out refused to send me cards without NFC, and as the NFC chip is integrated in to the chip-and-pin setup you can't simply destroy the chip as many Americans can (swipe isn't the usual way of paying around here)
More recently though one of the banks has wisened up and has sent me a non-NFC card, the other one is still NFC enabled.
That said, I have modified my NFC card to significantly reduce it's effectiveness, I scored the edge of the card near the chip deeply enough to break the antenna wire that runs around the periphery of the card. I know I can't make it detect on any NFC pad anymore, so hopefully that makes it relatively secure.
As for people suggesting Faraday cage wallets and such, I'm unconvinced. A proper Faraday cage has to have no gaps, and most of these are not that tightly constructed. I would not be at all surprised if many of them provide only a feeling of security rather than actual security.