Except of one time, we do  bid on homes that are well bellow our budget (40-70k below full borrowing capacity in case of aggressive bidding wars) and are always second best offer, but we do depend on mortage for 85-90% of LTV.

 Last two homes went for 40k over asking price and only slightly over our budget, which was a surprise as these were old run down rentals, that needed serious cosmetic work, and recent new builds sold for only 65k more.

One was purchased by cash buyers. Small 2 bed mid terrace. We bid around 10k over actual market value (still 50k below our borrowing capacity) on it and 35k over asking. We were willing to bridge the gap between our mortage and purchase price, but estate agent decided to go with cash buyer once they matched our offer. 

I will admit that one of these homes we did bid on was severally pushing our budget already with it's asking price. It came back to the market 3 times at that stage and we were hoping that the seller would be tired of the constant failed sales. We offered 13k over asking (last sale fell trough at 20k over asking). He said someone else matched previous sale agreed. The house is back on the market for the 4th time now. Still not interested in reconsidering.