Is it possible for a pregnancy test to be wrong at 6 weeks pregnant?

Patient: Is it possible for a pregnancy test to be wrong at 6 weeks pregnant?

Symptoms: I have been getting dizzy, having bad acid reflux, heartburn, I’m super sleepy, when I wake up I feel like I am going to throw up as if its stuck. I have been getting really bad hot flashes and lower back pain. I took a first response test and it said negative. also my stomach is retaining fluid and I missed my last period and the one before that was super light and only lasted 3 days.

Doctor: