Zero Economic Profit and opportunity cost?
Theoretically it is possible but only if the non profit organization operates at a cost which is lower than all other existing as well as expected new players as in a free market anybody with a much lower cost can enter and offer a price lower than the NPO and still earns a profit. And what about other NPOs in the same field. If they have a lower cost they will certainly offer a lower price. Secondly the NPO should accepted by the consumers. Practically speaking the market has many more complexities such as existence of brands and the information that price suggests. A lower price means lower quality which may not be accepted by the consumers. For example I may start manufacturing and selling mobile handsets at zero profit but still I might not drive Apple out of the market. Also, the world is not much perfect. To start I would have to install a capacity that can produce at a cost lower than the existing players and then offer a lower price to drive them out.

But still you can have examples of apparent monopolies in markets say for example 'online information'. What do you think Wikimedia is ? It is an NPO.