Postselection in geometric complexity theory
I like to think this way too, and I link it with the $P=NP\cap co-NP$ conjecture. It is mentionned there or there (I am having a hard time finding pages about this conjecture as I can only refer to it with math symbols that Google does not like).

A problem is in NP when you can give a certificate for a "True" answer (example : a hamiltonian cycle for the TSP problem or a valid assignment for the SAT problem). It is in co-NP when you can give a certificate for a "False" answer (example : a "proof" that the SAT formula is not satisfiable, a bad cut in a graph that prevents the existence of a hamiltonian cycle, etc...)

Actually, those may be the "obstructions" you may refer too. And this conjecture is about saying : a problem which is in NP is polynomial when I know what the obstructions are.

Now, the "obstructions" can take many different forms. For the matching problem (which is polynomial) it can be a partition of the graph (see Tutte's characterization of graphs admitting a perfect matching), or it can be the certificate given by Farkas's Lemma for linear programming (and graph problems that can be reduced to it). It can actually be a great many things, and so I usually "use" this conjecture in one direction :
When I can find no obstructions, I do not deduce that my problem is NP-Hard. Some obstructions are really hard to find... But when I have a complicated polynomial algorithm for a problem, I am sometimes convinced that "there must be an understandable set of obstructions, which would be easier to understand that the complicated algorithm".

Well... My two cents :-)

Nathann