As Ron knows I am an "older" guy from Canada. When I was getting our laboratory CSA (Canadian Standard Association) certified bac in the 1970s, the technicians had to be certified by CSA for cylinders, slump and air but a Professional Engineer did not. I was told by the CSA rep that they would expect a Professional Engineer to be able to read and do what is expected or else not do it. As C-31 states, field "technicians" must be certified etc. but it does not state that Professional Engineers must be certified. I am not sure, anymore, if this is the case in the States or not - or even in Canada as I have been out for nearly 20 years.
With respect to levels of responsibility. Some have pointed out that since one firm has done the casting and a different one does the curing and breaking, the latter would rightly state that they had nothing to do with the casting of the cylinders. But on the other hand, cylinders are cast only to prove that the concrete has the "potential" strength as stated for the mix design - it in no way shape or form guarantees that the concrete in the structure has the strength. As others have hinted, the actual (not potential) concrete strength in the structure is dependent on curing, vibration techniques, segregation, climatic conditions and the like. This is a contractor attribution; so who is to blame?
In the end, the cylinders are only a measure to prove that the concrete can achieve - it is no guarantee; and when the strengths come out low, all suspect the concrete and those casting and breaking and very few suspect the lack of curing, lack of consolidation, and other contractor attributables. I had a chart one time (which if my grey cells permit me to remember I will try to find) that shows how much effect specific casting and constrution techniques can have on the strength of concrete.