You may see that there is some flexibility:
There are a lot of Banks out there, and those banks are managed by Human Beings: yes, it's true.
Often, the person at the Bank is dependent upon the people in the field to give them information. Sometimes that information is slanted, sometimes incomplete. Sometimes the Bank will tell the Listing Agent what to LIST the property for, and sometimes the LISTING AGENT will be free to do as they wish.
Sometimes the LISTING PRICE will be set low to attract multiple buyers, and sometimes it will be higher than the COMPs.
When the Listing Agent submits an offer to the Bank, they may include an updated set of COMPs so that the person at the Bank can make a determination. Sure the Bank wants to MAX their income, but they know that there is a limit. Specifically, the APPRAISAL has to cover the Offering Price.
This is where it all comes together: If the Appraisal does not cover the Offering Price, then the Bank will not be able to sell the house, and the Max Price has pretty much been determined.
So, you may offer $50,000 over the LISTING PRICE, but you won't be able to buy the house, unless you are making a a CASH OFFER. The Loan will not be covered.
The Key to getting to that point, is to have your Agent do a good CMA, so that you will know how much to offer.
Good luck and may God bless