I have never heard of this. What airline does not allow you to print as many boarding passes as you like?
Delta
No personal experience. Poster on, I think FT. You can print as many as you want but requesting an electronic BP supposedly prevents you from getting a paper BP. I'm not 100% sure but I think he said getting a paper BP stops you from then getting an electronic BP.
Delta's website says the e BP replaces your paper BP. It says if you didn't receive your BP they will resend it at the airport. They don't say anything about issuing a paper BP under those circumstances.
I'm not sure what they'd do if you told them you left your phone at home.