How can change in Gibbs free energy ever be positive?
You didn't do anything wrong here. This was a very good observation. This is one of the criteria of spontaneity, which I believe you haven't learnt yet.
As you correctly wrote, $\Delta G$ can only be negative or 0 for any reaction. This is actually a consequence of the second law of thermodynamics, which also goes on to say that in any spontaneous reaction the entropy change of the universe is always positive.
Essentially, the answer to your question heading is: in any spontaneous reaction, the Gibb's free energy change can never be positive. Since you took the conditions of the second law, which essentially explains spontaneity, you arrived at this result.