With any investment offer, you have to have a strategy in place.
Know what houses in close proximity are selling for in four months or less. Use comps that are within 10% square footage range and with similar amenities (parking, architecture, finished basement, etc). Look at comps that sold low to state your claim to the bank. And look at high comps for your ARV.
Know how much work the property needs. Take pictures if necessary to state your case on why the price needs to be lowered. Many realtors do not know how much repairs cost, with labor and materials.
La Tonya has a great point of highlighting what it will take you to sell the after product. Advertising, Listing Broker, holding costs, etc.
If trying to give a low offer, there needs to be other factors in your favor. Every property, lender, and realtor are different, you have to try your hand at cracking the combination. Like previously stated, a bank will decide what type of hit they can and will take. If the property is a gem, the BPO will reflect that and they are not likely to give it away. You have a better chance at getting a steep discount on a short sale if there has been little to no offers on a property; which proves to the bank that something is wrong with either the property and/or the price.
The biggest thing is knowing what you are willing to pay for the property. If the offer does not get accepted, than just walk away. But, if you send out too many offers that never get accepted, than maybe you need to reevaluate your strategy and pricing.
Good Luck!