Question Summary:
Can I name my expected twins with the following names?

Question Detail:

We are expecting twins within a month. We would like to name our boys:
Mohammed Abdur-Rahmaan Afeef 
Mohammed Abdur-Raheem Ateeb
Can we name the above? What are their meaning?

Answer :

In the Name of Allah, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
We are very pleased to hear of this exciting news.  Allah Ta῾ālā grant you healthy, pious, beautiful, clever and loving children.  Amīn.

Muhammad Abdur-Rahman Afeef is a very nice name.  Muhammad is the name of our beloved Prophet salallahu alaihi wasallam.  Abdur Rahmān means the slave of the Benovolent One.  Afeef refers to one who is chaste. 
Abdur-Rahīm means the slave of the Mericful One.  Ateeb means the one who is reprimanded.  It is not advisable to keep the name Ateeb.[i]

Although the above mentioned names are nice, it would be better to keep a shorter name.  You may keep Muhammad for one and another name for the other child.  Or you may keep Abdur-Rahmaan and Abdur-Raheem.  It will cause difficulty for the parents and the sons themselves by having long names.  Every time a form needs to be filled, one will have to write the whole first name and the surname.  The passport will have a long name in it.  Teachers at school will have difficulty calling the child all the time.  These are just some of the realities of having such a long name.  Thus, it seems advisable to select a shorter name.    
And Allah Ta’āla Knows Best
Mawlana Faraz Ibn Adam,
Student Darul Iftaa
UK
 
Checked and Approved by,
Mufti Husain Kadodia
www.daruliftaa.net