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 seminar 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.