A problem in $X$ is not $X$-complete if there are any other problems in $X$ which cannot be reduced to it. One straightforward (but possibly only effective on trivial examples) method would be proving your problem is also in some other complexity class $Y$ such that $Y \subset X$.
For example, if you want to show that your problem is not $EXPTIME$ complete, then it is sufficient to show that it is in $P$, since $P \subsetneq EXPTIME$. However, it if you wanted to show that a problem is not $NP$-complete, then it is not necessarily sufficient to show that it is in $P$, since it is not known whether or not $P = NP$.
