Showing that a problem in X is not X-Complete
Actually proving $X$ is not $PSPACE$-complete (under, say, polynomial-time reductions) would be extremely hard to do. 

If $P=PSPACE$, then all non-trivial (i.e., not $\varnothing$ and not $\Sigma^{\star}$) and infinite problems in $PSPACE$ are $PSPACE$-complete under polynomial-time reductions. Since the existential theory of the reals has this non-trivial and infinite property, proving that it isn't $PSPACE$-complete would imply $P \neq PSPACE$. (See the answer to this question on CSTheory.SE for a sketch of the proof.)