Is it possible to get a positive pregnancy test 4 days after sex or. I had to be pregnant from a previous encounter?

You would have to be pregnant from a previous encounter. It takes a while for the embryo to implant and have enough of the supporting hormones being produced to cause a positive result. Usually you have to wait for two weeks after a missed period, 4 weeks after you ovulate (assuming a 28 day cycle), to accurately test at home.