Earlier quoted context omitted.
No. I was under the impression that the same wallet can have an arbitrary number of addresses. https://en.bitcoin.it/wiki/Address seems to support this idea.
You can have multiple addresses per wallet but BTC is still stored in addresses, not wallets. So if you have some BTC in address A and none in address B then you can't use B to buy anything. If you transfer money from A to B that leaves a record.
Alternatively, could I use a web wallet to bounce coins around to obfuscate ownership?