Exactly. And I've found that more often than not that the "we MUST complete this by xx/xx/xxxx or ELSE" mentality stems from some arbitrary date promised for no good reason. The company wasn't going to lose any more money or expose themselves to potential liability. It was just an arbitrary date marked in an MS Project file somewhere.
In situations where deadlines were missed, it ended up being OKAY and nobody really cared a week later. This then turned into a cycle where every deadline was tight and everybody got burned out because you can be sure that a manager who is bad at managing dates once will be bad at managing dates forever.
It seems that a lot of people in management lack good, common sense when managing workloads and deadlines.
I worked at a company for a while where the solution to any deadline was either to work longer hours or to throw more people at it. (They called it "swarming"). I'm pretty sure not a single executive there had read the mythical man month, or they would have realized how insane that was. You can't just brute force problems that require creativity.