What's the most you can bid down on a house? As much as you want. I've know investors who've bought houses for 60 cents on the dollar. A house is worth $400,000; they've bought for $240,000. Or less. And those aren't short sales or foreclosures. They're motivated sellers who want to sell quickly for cash. There aren't a lot of those deals out there, but there are some. And the point is: There's no "floor"--no reverse cap of 10%, or 20%, or whatever.
Your second question is "What's the lowest reasonable amount?" Well, what's "reasonable"? If you told me: "A reasonable offer is one that the seller accepts or counters 90% of the time" (I just made that definition up, by the way), then depending on the market you're in, it might be anywhere from 2%-6% off the list price. But I don't like list prices; they may be higher or lower than the actual value. What you should be aiming at is buying a house at or below what the comps suggest it's worth. So, determine the house's value via comps, then consider coming in somewhat under that.
Hope that helps.
My question to you is what is the Fair Market Value of the house today? You need to understand this first. Let's say that due to a short sale or other similar reason, the property is priced well under current Market Value. You are in luck! If it is a great deal at $350k, I would not hesitate to pay that price. Why mess around and risk losig the property? In the long term it might be a great investment and make you money.
If it is priced above Fair Market, then a rule of thumb today is that your initial offer should be about 10% below asking. Sometimes this will turn off the seller, but sometimes you will be surprised at the response. Understand that if they do not counter, you have to come up in price if you really want the house.
The last thing to consider is what is your motivation for buying? I sell distressed houses on occaision and I have always found that owner occupants will pay more than investors. Sometimes a lot more. So if you are an investor looking to do a flip you can be at a disadvantage offer price wise compared to a family purchasing.
Bottom line, 90% of the time start below ask, but know your facts first and do not be greedy.