Independant appraisals can sometimes still come back higher than what a banks appraiser will give (each person is going to use their own set of comparable properties, with the bank being the most cautious of the bunch). Your bank will only lend on THEIR appraisal, so keep that in mind.. Even if you end up offering more, you may wind up needing to renegotiate. Your agent will write an appraisal contingency in to your contract, I would hope.
A banks appraiser will typically use properties a half mile radius from the home, going back 6 months (sometimes more if needed). Is that what the current appraiser is using?
Feel free to call or email me with any further questions.
Keller Williams Langhorne
215 740 5643