One of my favorite explanation of the cohomology groups of low degree is that they arise as the automorphism group, tangent space and obstruction space (or where the obstruction lives) of a certain deformation problem.

My question is, is it possible to reverse the process of going from deformation problem to cohomology groups? Say I have H^0, H^1 and H^2 of a certain sheaf, is it possible to find a deformation problem such that these groups control, i.e. such that they arises as the Automorphism-Tangent-Obstruction of that deformation problem?

To be more concrete, say we have a smooth scheme X over k, and our sheaf is the tangent sheaf T_X of X, how do I "find" the problem of "smooth deformation of X"?

(I know I have been pretty vague. I know how to interpret cohomology classes as gerbes/torsors, but I'm not quite satisfied (I like these interpretations though). I want a way to find deformation problem that produces gerbes as obstructions...)

Quote from wikipedia: Reverse engineering is the process of discovering the technological principles of a human made device, object or system through analysis of its structure, function and operation. [I hope my usage of this phrase is correct.]