Consultant for well over a decade now. I've worked with quite a few MBB kids over the years as well as with clients who previously hired them. I'd say it's fairly uncommon for clients to actually yell at the analysts on the ground, but it does happen. It's more likely that the MBB associate or engagement manager gets yelled at for various reasons. But it's a lot more common for the analysts to be told "do whatever the client wants" and end up working absolutely stupid hours to cover client demands that should have been pushed back on, on top of internal commitments to stay on the right side of the "up or out" funnel.
So clients actually shout at consultants? Guess somebody's parents didn't raise them right.
I never got yelled at in consulting, and honestly if the client is upset that’s the partner’s fault. Most of the time there wasn’t a lot of pushback because the partner knew what the client cared about. The exception was PE firms who would push back on certain assumptions.
When I was in PE, I was the 25 year old shouting at some 45 year old CEO (with the license of my firm's MD) for BSing around. Fun times, but that was a very exceptional circumstance.