Has the dealership/repair shop ever repaired something but broke seomthing else?

Well, I'm having a small issue. The air coming into the car cabin through the vents is super HOT even on a cold day. I don't have the heat on. The ac is just running fine from the engine side but it's just not getting into the cabin. I posted a question on here about it and a lot of people said it's probably behind the dash, that the controls are messed up. Well, that got me thinking how could something like that have happened and I thought maybe it's just the age of my car (2002) but then I remembered that the air failed with in a few days of getting it back from the dealership for replacement of my clutch. Upon getting it back the dash wasn't put in correctly so it's loose and even the airbag on my stering wheel wasn't put on right (I have yet to touch either). Well, that showed that they were behind the dash doing something. So I'm going to try to take my car into a different dealership only because I'm a state away and see what they say. But, do you think they would pay for the repairs if it seems like it was due to the other dealers fault?