Pregnancy tests can not false positive.
Yes they can.
At-home pregnancy tests measure the amount of a specific chemical in the urine that determines if it is positive or negative.
Urine tests have never proven to be nearly as accurate as people believe them to be. Those little sticks you piss on can easily give you a false-positive, or even a false-negative.
In fact, using them in the first trimester grants the trickiest results, as the woman's body may not have produced enough of the chemical in the urine for it to go positive, producing a false-negative (or you may not actually be pregnant at all), or the body mistakenly produced the chemical while the woman wasn't even pregnant (this is usually caused by pseudo-pregnancy), giving a false-positive. That can also be caused by the tester being faulty.
Blood tests however, very very very rarely ever give false-results. Within the first trimester (first 3 months of pregnancy), blood tests are proven most accurate. By the second trimester (4-6 months), blood tests aren't needed, because by then you should be able to physically tell whether or not you're pregnant.