They may be able to still bill you and you may be legally obligated to pay, you are just taking away their way of automatically collecting the bill. You may still actually owe the bill; just because you didn't pay a bill doesn't mean you don't owe the money.
All this depends on the company, what you signed up for, if it was a contract, the TOS, etc, etc, etc.
For a real silly example to illustrate - lets say I signed up for Comcast and gave them a single use credit card number for automatic bill pay. When the second month comes around they attempt to charge my credit card and the card is declined. That doesn't mean I suddenly don't have to pay my Comcast bill, it just means they can't collect it automatically. Comcast will take a few months to cut off service[1] so you'll end up owing them several hundred dollars. Eventually if you don't pay they could send you to collections. Collections can take you to court and then when they win they can garish your wages, etc, etc.
Yes, I know this is a silly example and its unlikely to happen with the majority of "free trial" services on the net but that doesn't mean its responsible to basically advertise "yeah, just give any company a temporary number for a free trial and forget about it." Especially since a lot of services with a free trial are with companies that have a lot of resources.
A less silly example would be if I signed up for a gym with a two year agreement and a year later I canceled my credit card and stopped going to the gym. In that case the odds of me being billed further and sent to collections is very very high.
[1] they do that around here, I know someone who only pays their Comcast bill every 4 months or so...