TITLE: What techniques exist to show that a problem is not NP-complete?
QUESTION [16 upvotes]: The standard way to show that a problem is NP-complete is to show that another problem known to be NP-complete reduces to it.  That much is clear.  Given a problem in NP, what's known about how to show that it is not NP-complete?  
(My real question is likely to be inappropriate for this site for one or more reasons; I'm curious about what a proof that factoring isn't NP-complete would look like.)

REPLY [19 votes]: Here's one more pointer. 
Ladner's theorem states that if $P \neq NP$ then there is a problem in $NP-P$ which is not $NP$-complete. 

Richard E. Ladner: On the Structure of Polynomial Time Reducibility. J. ACM 22(1): 155-171 (1975)

Unfortunately the known proofs of this result produce a fairly unnatural problem. The zombie-metaphorical explanation is you take the head and torso of an $NP$-complete set and surgically stick on arms and legs of a polynomial time computable set. (If that ain't unnatural, I don't know what is.)
See this writeup for two proofs.