This is a requirement to pass basic training in the Army (or at least, it WAS a requirement when I went through basic training).
Everyone starts out in real bad shape for Army standards, and they gauge you on how much you improve since you started, but there are some required numbers that you must meet, such as your average mile time on the run or completion time of the obstacles. One of these requirements was to do 100 pushups in a row with no rest in between.
I thought that I would never be able to do it, but they have you doing so much running and strength exercises that it wasn't impossible to accomplish by the end of training. It really is a mind over matter thing, because your arms start to feel like they are on fire after about 35-40 pushups, and around 80 or so you are sure that they are going to fall off.
It took us more than 7 weeks, but I'm sure you could at least make some pretty good progress if you really focus on that program. You'll be satisfied even if you don't reach 100, because you will be able to see how much progress you made, and that is supposed to motivate you to keep trying.