Yahoo used to offer that as an option on their auctions. It's designed to replicate more an "online" auction. I believe there are some other online auctions which also have this feature. Yet eBay's market share is massive. Methinks neither sellers not buyers actually like this feature.
There's actually some serious research at
http://vwl3-10.ww.uni-magdeburg.de/anmeldungen/konferenz_2002/data/85.pdf.submit entitled "An Experimental Analysis of Late-Bidding in Internet Auctions" looking at the relationship between early/late bidding and whether an auction is extended automatically. The authors note than on Amazon, which has automatic extension, there's much less late bidding than on eBay, which does not. I think their conclusion was that the Amazon model produces (slightly) lower prices than the eBay one, but it's very technical.