The quick answer to this question would be no but it would definitely vary depending on each individual. A lot of teenagers have trouble with keeping themselves financially managed and end up spending money and racking up a bill pretty quickly – use the first month of the cell phone bill as an example.
If you believe in your son or daughter and trust them with a credit card, you could set a low limit to start with and build up that trust. Keep in mind that your child would have to have an income stream to pay this back in the first place or else you are just throwing them free money.
If your teenage child can not actually afford to coop with their current life then how can they be responsible for a credit card bill? If your son or daughter has extra spending money from every pay check, than it would be another story.
If you are trying to decide whether or not to get your teen a credit card, the best thing is to not take the risk. The rewards do not outweigh the risk in this particular case.