If we are talking about something like a phone bill, where you may incur charges during billing period and pay at the end, then you are correct and the company can take you to court because you are in debt to them for services provided that you have not paid for.
If we are talking about subscriptions where you pay for a month of service up front, then there is no debt. If they fail in charging another month to the provided credit card, then they are under no obligation to renew your subscription. Assuming they do not, then you have not recieved any service that you did not pay for. Their only claim is that you promised to by another month; however this claim is weak because you could have properly canceled your subscription before buying the next month anyway (and, in most cases, you would still get access until the period you had paid for anyway).
FYI: contracts can specify a lot of different payment terms. The term that you apparently desire to be in contracts you sign is against the interests of your counterparty. Most of them will have asked a lawyer how to write their contract. Their lawyers will often draft that contract to protect their client's interests. You will find that lawyers are rather good at doing this and are about as likely to be out-thought by a programmer's parsing of the English language as you are to be out-Fizzbuzzed by a randomly chosen attorney.
Again, if you have agreed to their terms and conditions, then you are under obligation to abide by them (within reason; there are limits on what can go into a T&C). The T&C almost certainly obliges you to pay them. Can anyone cite precedent for failure to pay magically releasing someone from that agreement?
Gizmo is cleary referring to services where you pay upfront for, e.g., one month of service with some form of auto-renewal. He's not talking about failing to pay for services rendered in advance of payment.
If we are talking about subscriptions where you pay for a month of service up front, then there is no debt. If they fail in charging another month to the provided credit card, then they are under no obligation to renew your subscription. Assuming they do not, then you have not recieved any service that you did not pay for. Their only claim is that you promised to by another month; however this claim is weak because you could have properly canceled your subscription before buying the next month anyway (and, in most cases, you would still get access until the period you had paid for anyway).