The pumping lemma for regular languages states, that one should consider a string $w = xyz, w\in L$, that is, every possible division of $w$ into $xyz$.
The article on wikipedia says, that this version is a special case of a more general version, in which one should consider strings $uwv = uxyzv$.
When using that general version of the lemma, does it require considering every possible partition of the string into $uxyzv$? I.e. does it require considering every possible $u$ and $v$?