I currently work for a major EPC consultant in the oil & gas industry. We are ISO 9001 registered, and have committed to checking all calcs. This is actually too much. Some stuff needs checking, maybe even more than once, other items could probably go with no checks.
In my earlier commercial consulting days little was formally checked, but I used to do my own reality checks. Manual load take downs were checked (reactions totalled, applied loads totalled - were they reasonably matched?). Larger more critical members, I would recheck, with a different method if I could.
As someone earlier mentioned even senior engineers can make silly little mistakes, and as one of my profs used to say (his Law 199) no one mistake usually causes a collapse, it takes an accumulation of mistakes. An early boss tought me a simple rule - does it look right? Many mistakes can be spotted by looking at something with a detached eye, and the absurdity can jump right out at you. Experienced designers (technicians) and draughtsmen can often spot these goofs.
In the calc checking that we currently have to do we have two options. Literal detail calculation checks, or alternate approach. I generally hate the detail calc check, particularly with some individuals whose calcs while maybe correct are extremely difficult to follow, so I tend to favour trying to come at a problem from a different angle and see if I arrive at a similar solution whenever I can.
On one project with an intermediate engineer whose calculations were always difficult to follow, I was getting quite frustrated trying to follow/check his numbers, so I decided to sit back and look at the problem a little differently, and the flaw that I finally found in his calculations was not in the number crunching itself, but in an assumption of flange continuity that I was not convinced was realistic (vessel lifting beams were cantilevered off eiher side of a deeper saddle extension, so that the top, compression flange was discontinuous, with only a partially stiffened web plate connecting them). In lieu of doing a detailed finite element analysis to make sure that we would not get local plate buckling somewhere in the beam/saddle combination causing us to drop the 150 ton vessel/platform assembly during the critical installation lift, I decided to add some temporary lift bracing between the upper and lower top flanges to transmit these forces via truss action. Although the other end was more lightly loaded and did not present a "failure" problem in the number crunching, higher than anticipated deflection during a lift to weigh the module caused us to decide to add the flange braces at that end too. This was partially exacerbated by the fact that the vendor had given us a bad c-of-g and so we actually had more load on the other support than we had expected. But basically the check here had been sit back and look at the overall concept, not get tangled up in all the detailed numbers.