Conservation of angular momentum and its prediction by Newtonian Physics 
The situation, in brief, is that newtonian physics is incapable of
  predicting  conservation  of  angular  momentum,  but  no  isolated 
  system  has  yet  been  encountered   experimentally   for   which
  angular   momentum   is   not  conserved.  We  conclude  that 
  conservation  of  angular  momentum  is  an  independent  physical 
  law,  and  until  a  contradiction  is  observed,  our  physical
  understanding must be guided by it.
Dan Kleppner

Why Is newtonian physics incapable of predicting it? Isn't it a consequence of Newton's laws of dynamics?
 A: You are quoting out of context. 
The quote you present is from page 307, the following is from page 306

The situation shown in figure (a) corresponds to the case of central
  forces, and we conclude that the conservation of angular momnetum
  following from Newton's laws in the case of central force motion.
  However, Newton's laws do not explicitly require forces to be central.
  We must conclude that Newton's laws have no direct bearing on whether
  or not angular momentum of an isolates system is conserved, since
  these laws do not in themselves exclude the situation shown in figure
  (b)

Newtonian mechanics comes with the implicit assumption that the effect of any force is independent of spatial orientation. If you grant that assumption then conservation of angular momentum is a logical consequence.
Obviously, if you decline granting one of the fundamental assumptions of newtonian mechanics then you have moved to a realm where newtonian mechanics may not apply.
As I understand it, Kleppner is arguing that we should allow for the possibility that the Newtonian assumption that implies conservation of angular momentum may not hold good. 
Kleppner does assert that within its realm of applicability newtonian mechanics implies conservation of angular momentum.
