If they cared, it would be handled or they wouldn't use consultants.
This is where providing transparency and insight (in a "how can we fix this?" way) up to management can help engagements go more smoothly. If everyone was aware from the first that "that QWSV system access issue" has been impacting the projects for weeks, that's much better than "the project is weeks behind" being communicated up at a random status meeting.
I wouldn't advise to participate in the blame game or to point fingers. You are not here to cause troubles and trouble is bad for repeat business.
P.S. Organizations are dysfunctional and the majority of consulting time is wasted. Lesson one of being a consultant is to live with it. :D
But if shit really hits the fan, at some point management will be asking "Who was supposed to do this?" And between that and the finger possibily being pointed at you... soft skills in handling politically tense situations well are useful.