My last period ended on the 4th of June. All of last week I've had brown spotting, and just recently its started to appear redish. Mid last week I did a pregnancy test and it was possitive. Could there be a possibility the outcome of the pregnancy test was wrong?

It could be, but a false negative is far more likely then a false positive. The odds that it was a false positive isn't high.

Have you tried testing again, or get a blood test to confirm?

Chances are you are pregnant. False positives are rare. Time to make an appointment with an OB.