No. Non-profit status confers significant tax benefits that for-profit businesses cannot take advantage of. Therefore, it's a tightly controlled status.
A non-profit company can certainly make money, even lots of money, but it has no stock and no investors and cannot pay dividends like a for-profit company. It can pay salaries and expenses, of course. However, a non-profit must fall into one or more of a list of allowable purposes: educational, religious, charitable, scientific, etc. The money it brings in must be used to further its mission under those categories.
If you're just creating an "ordinary" business that makes and sells widgets, it must be organized as a for-profit business, and pay taxes on those profits, like every other widget maker.
For those reasons, a business cannot start as a non-profit and then transition to a for-profit model. You would have to shut down the non-profit, sell off the assets for cash (or donate them to another non-profit), and then donate the cash to another non-profit. Otherwise a start-up business could be a non-profit, enjoying tax breaks and running on donations, until it figures out how to make a real profit, and then switch and, well, profit.
You can of course start a for-profit school, museum, hospital, etc.; they aren't required to be non-profits. But NGOs and other donors will stay away, and you would have to be a for-profit business from the beginning.
This is specific to the U.S., but other countries have similar rules.