I was able to get to 20 pushups and I kept increasing by one or two a day. Then when I hit around 35 I pushed myself and keep going until I hit 40. It took a little over a week and a half I think.
Not the OP, but the method that worked for me when I was training for the military years ago was to (situation permitting) drop down and do five push ups every half hour for most of the day. Do that for a week, then the next week add five more. In a few weeks you’re doing 20-30 push ups every half hour and the maximum you become capable of doing if you have to take a fitness test goes through the roof.
This technique is called "grease the groove" and it's useful anytime you've plateaued on reps for an exercise.
Edit: also focus on your form!
Has anyone tried it? I believe you want to try for 50% of maximum.
One method is doing sets of 10 reps (or whatever number of pushups you can do in a row right now) and gradually reduce the time between sets. For instance, you start resting 60 seconds between sets. After a week, you rest 40 seconds between sets. And so on and so forth. After a few of months you should be able to do 40 pushups in a row.