No limit to bandwidth means that it would be possible to send any amount of information across a network with no latency. Such an achievement would change more things than just the web. For instance, with the capability of limitless bandwidth, data storage and processing power would no doubt have made equivalent leaps as well. These are components of networking infrastructure as well as general purpose computers. So, networking equipment that provided limitless bandwidth would also include processing power to handle the load - processing power with no limits. Moving any amount of information with no latency also means you need some place to put it - data storage with no limits.

With these limits removed, there might be no need for a web at all. The ability to move any amount of information instantly might mean we keep a copy for ourselves of everything we interact with, continually accumulating and indexing data at a constant rate from other information providers for the rest of our lives. From this I can imagine having my own reference database of accumulated information that becomes our private web, or life encyclopedia.