can be understood as:

born in Bohemia - and all our puppies are and will be born in Bohemia

born as Bohemian (by name) - our surname is Cech, what means Czech or Bohemian

 - and our puppies are and will be members of our family, so they are (will be) Bohemian twice

Bohemian by nature - free in mind, natural, wild... - what I think a Tibby is...

can be shorted to BBB, as made of three words of the same letter

 - and I really like such names