Physical Education

I'm in IT, so generally when it comes to physical activity I'm pretty stagnant. Recently, I took it upon myself to start doing some running alongside a number of push ups and to slowly work my way up. But here's my problem: I've been reading that it's best to space out push ups so that your body gets some time to heal, since apparently it's valuable in order for the muscles to grow. So I've been doing them ever 2nd day, which has worked relatively okay so far. The first few weeks were grueling because I lacked a lot of the stamina to do them, and could only do about 5 at a time. Now I'm up to two sets of 25 push ups per day, but what worries me is this sensation that I'm plateauing. Once I reach 25, I have to stop, wait a few minutes, before I do the next 25, which is fine, but if I bother to push for a few more, let's say, 28 or 30 push ups per set, my arms literally give out.

The good thing is they don't feel sore, but it's like my arms just physically cannot go any further. Does anyone else have this problem? How did you get around it?