Great question Karen! It really depends on the house and how well it is initially priced. If a property is properly priced, you MAY only get one chance. When the seller gets an offer on their property they have a few choices. They can accept, reject, or reject with counter-offer.
Now if you are not the only offer on the property, this can affect your negotiating strategy a lot. The seller still has the option to just pick one offer and accept it. There are no rules stating that they have to give anyone a second chance. Most often, however, in a multiple offer situation, the seller will ask all parties for their "highest and best offer". This is your opportunity to improve your offer and set yourself in the best possible position for the seller to accept your offer.
Please understand, however, that the highest price does not ALWAYS win out. The agent will assist the seller to evaluate the buyer's capability of completing the purchase. They will look at percentage of down-payment, type of mortgage, inspection clauses, pre-qualification letter or proof of funds letter, etc. I still remember one property I marketed where the bank/owner chose to take a $155,000 cash offer/close in two weeks, and left four higher offers (one was $180,000) on the table because, even though they were over asking price, they were looking for mortgage financing with less than 10% down.
Ultimately, it is the seller's choice as to how the negotiations are handled. They have NO OBLIGATION to go back and forth with a buyer, and can accept an offer at anytime. That means that if the buyer insists on small steps in the negotiations, they have opened the window for another buyer. For instance, you may have been negotiating with a seller for a day or two (or more), and maybe you are only a small distance apart from a fully accepted contract. Another buyer could present a strong offer and the seller has every right to abandon negotiations with you and to flat out accept this other offer without negotiations, without notifying you that another offer even exists.
It's important to be represented by an agent with good negotiating skills. If you have not hired an agent yet in your area, I would be glad to provide you with a quality referral. With 26 years selling Connecticut real estate, I have had the opportunity to work with several excellent agents in your area. I would be pleased to be of assistance. Feel free to call me at 860-688-1400/ ask for Lisa Orme.
Good luck Karen!