In my case, the most unrealistic deadline was when I was put on a project for 30 person days in 2008. The project had been running for about 6 months at that point.

I spoke to the project manager about my tasks and she told me to finish the fat client. So I immersed myself in the sources. And I was horrified to realize that not only was it not even a POC, but the performance was lousy to say the least. It took about 70 (sic!) seconds to start the program, read in about 20 records from a database and display them as a hierarchical structure.

I asked the PM when I was supposed to have finished my work, and her response was, "Yesterday."
"Very funny," I replied.
"No, really," she said, "the deadline was yesterday."

It took me an afternoon to speed up the fat client startup to 6 seconds. And then it took us another two weeks or so to identify the processes in discussions with the technical project manager. Because that didn't exist yet either.

About 1.5 years after the deadline, the software system - consisting of the fat client, mainframe modules and purchased software - was stable enough to be rolled out.