After reading the recent question "Is the complement of $\{ www \mid ...\}$ context-free?"; I remembered a similar problem I wasn't able to disprove:
Is $L = \{ ww' \mid w,w' \in \{0,1\}^* \land |w|=|w'| \land HamDist(w,w')>1 \}$ context free?
Here we require that the two strings differ in at least two positions (the Hamming distance must be greater than $1$).
It is context-free if we require that $HamDist(w,w')\geq 1$ (i.e. the two strings must simply be different).
I suspect that the language is not context-free: if we intersect it with the regular $0^*10^*10^*10^*$ we get cases in which a PDA should "remember" two positions in reverse order after reaching the half of the string.
Update: if we intersect $L$ with the regular $R = \{ 0^*10^*10^*10^* \}$ we get a context-free language as showed by domotorp in his answer; a slightly more complex $L \cap R' $ with $R' = \{ 0^*10^*10^*10^*10^*10^* \}$ (one more $1$ to "keep track" of) still suggest that $L$ should not be context-free.