When to pump up and down?
To fix symbols, let us choose the decomposition $z = uvwxy$.

One of the examples where $|vwx|\le p$ comes in, is the language $L =\{ a^mb^na^mb^n\mid m,n\ge 1\}$.

Start with the "generic" string $a^pb^pa^pb^p$ where $p$ is the pumping constant. To have any chance of finding a pumping pair $v,x$ such that $uv^iwx^iy$ is within the language is by choosing $v$ in the first set of $a$'s and $x$ in the second set of $a$'s (or similarly for the $b$'s). Now the length of $vwx$ will be more than $p$ as it contains all $b$'s between the two sets of $a$'s. 

Without that restriction $L$ can be pumped, and the pumping lemma is no longer suited for proving $L$ is not context-free.