I had sex 3 days before I came on my period, 2 weeks later the hospital tells me I'm pregnant could I have gotten pregnant around that time or before?

Yes, you could have gotten pregnant then or before. Sometimes one may have the at least the first part of the next period.

There is no way to predict an exact date of conception, but they will get fairly close latter as to when to expect the baby give or take a little bit.

The best will be give or take a week or even two weeks on guessing time of conception.