Here is what both    Feynman,  Grothendieck (and my father)  said: have several   projects in mind at all times. 

Grothendieck   explains  a two year  state of depression he went through at the beginning of his career  to the fact  he single mindedly followed one goal which turned out to be very illusive. 

Feynman explained his success  on  the fact that the he always  had several questions in his mind  and  kept an open eye  for anyting that might relate to this. That is why  he found attending seminars so much more helpful.

As for successful  PhDs dissertations I read somewhere, long ago,  that there are of two types

- the type where you find a new method for an old question,

and 

- the type where you find a new  question for an old method.

Statistically, the 2nd type is more prevalent.   Obviously that is a rough classification and  dissertations are  cocktails  of both types.

So to answer your question, should you  try to solve hard problems, my answer is yes, but remember that, even if you do not get the whole dinosaur   in you dissertation,  his tail may be  good enough to  make you a Doctor in Sciences.