I had one letter writer only submit a letter to half of my schools. I sent them a few polite reminders and never heard back, so I decided not to apply to three programs that required letters by the application deadline. The fourth school accepted LoRs after the deadline so I applied anyway. I just received a provisional acceptance from the program, but I'm worried that not having a third letter may still end up harming me.
The professor has always been extremely helpful in the past though, so perhaps something came up or things just got busy. I had a few professors turn me down junior year when I needed help establishing an internship program at my university. I really just needed a sponsor, but they were busy taking care of sick and elderly parents--which I completely understand. I barely finished my applications before my own grandmother became severely ill. Things happen. It sucks for both sides.