The prior answers are accurate but please do not take any one opinion you receive here as "THE" correct one. Every real estate transaction is unique and should be treated as such. Hopefully you have a real estate agent working for you and will help you determine what is a reasonable offer based on the exact same comparables that an appraiser will be using.
Banks are indeed scrutinizing every dollar they lend. They are also holding the appraisers accountable for every detail of their appraisal. This does not mean that it is the appraiser's fault if the appraisal comes in for less than the agreed-upon sale price. It could possibly mean that the house is selling for more than the most RECENT comparable sales.
Recently, I have had two transactions in which the appraisal came in lower than the agreed-upon sale price.
In the first situation, the seller came down on his price....but that was only because it went under agreement within three days of going on the market...and for above HIS asking price.
With the other situation, the buyer had to come up with a larger down payment because the seller would not lower his price...and the buyer definitely wanted the house.
The best advice I can give you is to be prepared for anything! In this market, especially here in Massachusetts (where the market is like the weather) what applies to one may not apply to another!