Which are the cases where the "pumping lemma" doesn't work?
The standard formulations of the regular pumping lemma all force the pumped part to be at a specific place -usually the beginning of the word- and of a very restricted length. Thus you can trick the lemma by contructing a language that allows just about anything in the first part of the string and gets complicated in the second part. Take for example:

$$\{wba^nb^n\mid w\in\{a,b\}^+,\ n&gt;0\}\,.$$
Since the pumping lemma just states the existence of its constants, you can pump anything in the first part of the language without leaving it. However, due to the last component the language is not regular.