What information, if any, can a serial number provide about a bike?

Depends on the bike, often not much.

Sometimes serial numbers can be deciphered. Sometimes they're purely sequential, other times they use a system that includes dating and/or sizing information. They can also incorporate letters in addition to numbers that can have meaning. To give but one example, Raleighs that have an "SB" serial number prefix are upper-level frames crafted at Raleigh's famous Special Bicycle Developments Unit at Ilkeston. Therefore, one of the first questions to ask upon spotting a frame or bike in the famous Ti Raleigh red-black-yellow livery is, "What's the serial number?" I was just talking to someone yesterday about the system used on the 7-Eleven team bikes, and I believe we've learned something new about dating those. Learning serial number systems and their meaning goes on and on - though sometimes there isn't much meaning there.