I could be wrong, but I believe there is a minor error in 20.16. It says that renice +3 13848 will increment niceness by +3, but it appears to set niceness to 3.
It seems that you are right. In fact, if you run the command again (using a correct pid) it will keep the priority. I think that the person who wrote the document was thinking in the moment that you run the command for first time, as in fact it will increase the niceness from 0 to 3.
I'm reporting it now, thanks for telling us.