I was in the Marines. When I went to boot camp, I was lucky to do 10, I could probably do about 7 or 8.
3 months later, 20 wasn't a problem. Matter of fact, I just bought a pull-up bar a few weeks ago, and I was still able to do 20, even after getting out way back in 96. I do a lot of bicep work, but hadn’t done a pull-up in years. Not to mention, I'm about 40 to 50 pounds heavier now.
If you work at them every day, you could do 20 in a few months, if not sooner.