Bad Faith involves the adoption of a principled posture which is unprincipled. Other, more simple descriptors may be used to label such a posture - "lying hypocrite" comes to mind - but bad faith is a term which is worthwhile advancing for its precision of meaning, and emotional coolness. (Calling someone a lying hypocrite can be provocative, whereas claiming that he or she is acting in bad faith may be, or ought to be more tolerable.)
Here are some examples of bad faith to both clarify the concept, and to illustrate its utility in identifying and challenging humbug.
The first example is almost literal, and comes from my dim recollection of an episode of Yes Prime Minister. Jim Hacker is conferring with Sir Humphrey Appleby about possible appointees to the newly vacant post of Archbishop of Canterbury. The bottom line is that Sir Humphrey makes a compelling case for appointing someone who doesn't believe in God. Their appointment process is itself in bad faith, but an agnostic or atheist Archbishop preaching (with apparent sincerity) from a pulpit about the resurrection and the life everlasting would also be acting in bad faith. (Well duh!)
Some less ecclesiastical examples may serve to illustrate the broad utility of the concept.
A hypocritical journalist acts in bad faith when he or she claims to be reporting news - while in reality he or she creates news by provoking newsworthy incidents.
A hypocritical social researcher acts in bad faith when he or she claims to be researching a topic in order to discover underlying reality - while in reality he or she discards and doesn't report results which don't support his or her cherished hypothesis.
A hypocritical peace activist acts in bad faith when he or she expresses public anguish at the death of non-combatants while privately delighting in such casualties - as civilian deaths add weight to his or her rhetorical position on armed conflict.