I wouldnt say it is bad but be careful of what you wish for.

On the flip side if someone is merely after fame that is not such a bad thing by comparison. It is not like those that seek wealth and power, it is at least going a more ethereal and potentially much less destructive goal. Having lots of fame doesn't necessarily take from others.

The problem with a lot of fame is that the more of it you get, the more decided people will be. A good modern example would be Taylor Swift. Some people love her, many don't even notice but some call her Taydolf Swifler.

Turns out you cannot get just the good, you have to have the bad come with it as well. Unless you are Weird Al, everybody loves him!