The more crowded your chickens are, the more likely you might experience problems like picking, pests, etc. Also, it depends a lot on your weather and your birds' daily routine (will they be able to go outside any time they want (day time), will they free-range outside of their fenced area, etc?
For instance, I had 8 chickens in a coop that size and everything was fine. My outdoor run area was 12' by 12' (144 sq. ft) and it became bare ground after about 2 - 3 weeks.
Now, I let them free-range whenever I am here and only use the penned area when I go away. It still becomes bare and I have to move it around every so often.
I'd say if your climate allows them to go outside every day (even in their fenced area), you could get away with 10-12 birds, or maybe a few more. This is especially true if they get to range and if you provide 3 or 4 nest boxes. You're doing the right thing by starting with a smaller number and then adding a few more.
Good luck!