Would banning anonymous comments reduce cyberbullying

Now picture this, imagine several high profile cases filed on an issue almost as anonymous as that of ‘cyberbullying’. This may not make perfect sense to you, however, allow me to explain further.
Cyberbullying has been defined as the harassment of computer users, often teens and preteens, through various forms of internet communications. Now the groups opposing this cause, which in this scenario is banning anonymous comments, speculate that the right to protect your identity online in other words remain anonymous is somewhat a human right and should not be ‘infringed’ upon ob account of the bad behaviour of a small group of people.
On the other hand, the groups backing the cause claim that the anonymous nature of the internet directly or in my opinion indirectly leads to such immoral behaviour. These people are willing to push further and by that I mean getting the government to grant laws that somewhat force internet access providers to verify and retain the ‘true’ identity of all users, making false identification put up on the net illegal.
Those are pretty high demands I would say but the cause is worth supporting to secure the psychological well being of those affected by all this..... What’s your opinion??