How is it possible for a company to show positive net income but go bankrupt?

Yes. Two examples include deterioration of working capital (i.e. increasing accounts receivable, lowering accounts payable), and financial shenanigans.

Financial shenanigans acts or actions designed to mask or misrepresent the true financial performance or actual financial position of a company or entity. Financial shenanigans can range from relatively minor infractions involving creative interpretation of accounting rules to outright fraud over many years.

The answer to this question will always be yes. No matter the situation, fraud exists in our society and a perfect example is Enron. What happened to Enron is why SOX was passed through legislation. Answer the question and then give an example, but there is not need to get hung up on the answer.