You really need to give more detail in order to have good answers to your question.
IE, what part of the country do you live in? In certain locales, 20,000 will obviously go much farther than others. I currently live in the Los Angeles area. If you were asking that here, the answer would be a resounding "NO". There is a high cost of living here. Even within states (California for example), costs DO vary. If I lived in say, central California or the Bakersfield area, or out near the desert somewhere here, I would pay less and thus I might be able to live more comfortably on that. I lived in San Antonio for three years while I was doing my teaching certification. If you asked the question there, the answer would be "probably" but you might struggle. San Antonio has one of the lowest costs of living in the country.
You havn't given us a lot of detail about your current bills---ie, do you pay a car note, what is your rent? what are your costs for car insurance, credit cards, etc?
Certainly, a lot of the things you are doing--sharing an apartment, taking public transportation,etc, will save you money. Other things that you can do include taking lunch to work (and not eating out), clipping coupons whenever possible, etc. You should look into whether or not you and your son qualify for some type of assistance--more than likely as a student, you do. Also, why are you not receiving child support?
The short answer is--'maybe', but that is a tight budget, and my 'maybe' is really more based on someone supporting just themselves--not a child, with no child support. Everybody's situation is different, and just because someone else can make it on that money, doesn't mean you can, especially if you do not receive any assistance and live in a high cost of living area. If you give a little more detail, I can provide more of an answer.