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Yes, non-profits can make a 'profit' it just can't distribute them to members / shareholders.
Here's how it happens - you ALSO have a business, that leases things to the nonprofit, or provides some other services. Leasing cars and buses, providing laundry services, etc. And of course you pay yourself a great salary.
Thats called self-dealing and it is illegal. I'm not saying it doesn't happen but its like Step 1 IRS agents look for when you're getting audited. Defending against "self-dealing" charges are so common that I know directors of non-profits who refuse to even do things like buy themselves a burger for lunch when travelling for work because defending against the claims of self dealing just aren't worth it.