I faced a similar problem with one of my professors.
I asked him whether he is unable to write the letter and if I should ask another professor. He told me I should call to remind him and I called him couple of times (had told him about the letter at least 2 months earlier). I also told him that my application would not be complete until he submits the letter.
Another one of my professor (a very good friend as well
) submitted the letter the night of the deadline (~11:00PM).
I guess considering their work schedules, they are hard pressed.