New printers often come with a trial of the manufacturer's subscription plan, which will automatically order ink when it's running low. I guess if you activate the subscription when it runs low, it will send a new cartridge straight away. This probably also works with a real card number if you just cancel the subscription before it starts charging.

I'd assume that only works once per printer - or at least it's enough of a pain to repeat it that most people would rather pay. The smallest subscription plans are pretty cheap.

That’s got to be it. No way would a test card number actually work. Live payment processor environments decline them. They only work in payment processor test environments.

If the merchant’s backend doesn’t special case them and sends them through, they will be declined.

In HP's case, the cartridges will not work if the subscription has been cancelled.