You can bid as low as you want...now whether the seller wants to sell it to you at that price, is another thing! :)
An offer on a properly priced home typically can be seen coming in at 5-7% below list price and ending up within 3-5% of list. If a home has been sitting for many months and hasn't gotten an offer in that range, then the next relevant step is for the seller to drop it by 8-10%. So, if the price of the home is not reduced to that amount, the seller isn't motivated to sell their property and probably won't accept anything lower or they would have done that.
Ultimately, it all rests on being educated about the market and a Realtor can help you with that.
Are you looking at a specific property? Also, are you only wanting to buy in Jeff Park or are you looking in Norwood Park as well? I've been watching both areas for NC homes for several months now and there are some to look at there.
It would depend on which home to which you are referring (You're not talking about the one on Melvina, are you? Because it's under contract... but with a house sale contingency).
There is new construction in Jefferson Park for between $500,000 to over a million... so it's difficult to say without knowing the specifics of the home, where it is located in Jefferson Park, how much other new construction is around it.
The only way to know "how low" you can go is to do very recent comparables to determine a good market value for the home. With new construction it can be more difficult since it's harder to find true comps, but an experienced Realtor will know how to adjust the values accordingly in order to formulate a fair offer.
Builders are sometimes willing to make concessions and discounts, depending upon how motivated they are, in order to move on the next project, but again, it's not safe to generalize.