Way too optimistic: in many abelian categories there are not enough projectives (and in the dual category there are not enough injectives). 

The most standard example is sheaves of abelian groups on a topological space X. For most X, this category does not have enough projectives. See for example this question where this was discussed: http://mathoverflow.net/questions/5378/when-are-there-enough-projective-sheaves-on-a-space-x/5470#5470

On the positive side: if A has enough projectives and I is a small category then the category of functors $A^I$ has enough projectives (assuming arbitrary sums exist in A). In particular, the category of complexes in A has enough projectives. See this question: http://mathoverflow.net/questions/6776/how-to-construct-pair-of-adjoint-functors-from-category-a-to-category-adcategor/6836#6836