If you're looking at a home that has sat on the market for months or years, it is quite likely that the seller has priced it above market value, and even if they received an offer the bank would require an appraisal, and they would need to renegotiate the price.
I'm advising my buyers to seriously look at homes that have been on the market for more than 30 days and are priced about 20% over their target price. Smart Lowballs can work! If you don't have an agent, give me a call if you'd like to discuss a specific property. I'd be happy to help you run an analysis on what the "right" price should be.
Assessments have nothing to do with funding. You may mean Appraisals? If the appraisal comes in lower than the agreed upon purchase price, the buyer can pay the difference between appraised value and purchase price (unless it's a VA loan as the buyer is not allowed to pay the difference), the buyer and seller can agree to 'split' the difference, the seller can agree to lower the purchase price to appraised value, or the contract can be cancelled if buyer and seller cannot agree (if there is an appraisal contingency in effect).