I expect you can't market goods with that name, but nothing to stop you calling yourself Harper Beckham Caran.
Piers Morgan does have a point, though: can they demand privacy for their children while also trademarking them? Would the Mail be able to run a story about Harper being a naughty girl at school, as long as they printed it on the business pages and said the brand was being put at tisk?
I am sure that Mamy is right @ 22:15, but if they had not done so..........do you think that "someone" else outside the immediate family would have done so? Perhaps someone who you would not want your children "linked" to, however tenuously.
Peterborough united had been known as "the posh" since the 1920s. then Victoria beckham trademarked the name and promptly took peterborough united to court. she won. it would not surprise if this move led to court cases involving others who've had beckham-related names for more years than the beckhams have been alive.