TITLE: What determines a model structure?
QUESTION [18 upvotes]: It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).

cofibrations and weak equivalences
fibrations and weak equivalences

The second statement follows immediately from the first by duality.
What about the following classes of maps/objects (A short argument would be very helpful)?

cofibrations and fibrations
cofibrant objects and weak equivalences
cofibrant objects and fibrations
cofibrant objects and fibrant objects

I think each of these classes determine the structure respectively. For the last one I suppose that one has to use framings but I cannot see how to do it.
Edit: Thank you all for the illuminative answers.

true
?
true
false

REPLY [15 votes]: Mark's answer explains why (1) cofibrations and fibrations do determine the model structure, and Charles' and Tom's examples show that (4) cofibrant objects and fibrant objects do not.
For (2), any weak factorization system (L,R) on a category determines a model structure in which L is the cofibrations, R is the fibrations, and all maps are weak equivalences.  Thus, it suffices to find a category C admitting two wfs (L,R) and (L',R') with the same cofibrant objects, which is pretty easy; take for instance C=Set with (all functions, isos) and (monos, epis).
Finally, for (3) it is true, surprisingly, that the cofibrant objects and the fibrations determine a model structure.  In fact, merely the cofibrant+fibrant objects together with the fibrations also determine the model structure.  This observation is due to Joyal and can be found on his Catlab (currently Proposition 4).