Is $L=\{ xyx^Ry^R \mid x,y \text{ is an element of }\{0,1\}^*\}$ context-free?
No, it is not a context-free language. From my original SO answer: consider the string $0^n1^n0^n1^n \in L$, where $n$ is the pumping length from the pumping lemma for context-free languages. This fails the pumping lemma for context-free languages; in particular, we have seven cases for $vxy$:


$vxy$ consists entirely of 0s from the first part of the string. Pumping this up or down will cause the numbers of 0s in the first and third parts of the strings to be different, so we're no longer in $L$.
$vxy$ consists entirely of 1s from the second part of the string. Same problem between the second and fourth parts as between the first and third parts mentioned in case 1.
$vxy$ consists entirely of 0s from the third part of the string. Same but opposite problem as in case 1.
$vxy$ consists entirely of 0s from the fourth part of the string. Same but opposite problem as in case 2.
$vxy$ straddles the first and second parts of the string, beginning with 0s and ending with 1s. Pumping will change the the numbers of 0s and 1s in the first and second parts of the string, and they won't match the third and fourth parts of the string anymore, so we're not in $L$ anymore.
$vxy$ straddles the second and third parts of the string, beginning with 1s and ending with 0s. We have the same problem as in case 5.
$vxy$ straddles the third and fourth parts of the string, beginning with 0s and ending with 1s. We have the same but opposite problem as in case 5.