My reason for asking this is I am developing a product for deployment in the Third World that Bitcoin is a great fit for, but it must be able to function with only intermittent access to the blockchain and support 50-100 users.
Well, just to clarify ... if the users are running nodes and are connected to each other (e.g. on a LAN), then only one node needs to get blocks from the main net, and can peer with the remaining ones locally.
The bandwidth to receive the blockchain is about 100 MB per day. Dialup internet provides way more than that even.
Mining requires continuous communications. Mining doesn't require much bandwidth, (the same 100 MB per day would be plenty) but the worker continuously needs to be getting new work from the pool as mining is the process of verifying transactions and that batch of transactions can change second-to-second. So if there's no continuous communications then there's no mining.
With your scenario of once-a-week connectivity, that would only work if all parties could trust each other that there was no double spending until connectivity is re-established. Thus the nodes on the desert island would be stuck with whatever block was last received. They can use Bitcoin just fine, just that the transactions will stay at 0/unconfirmed until the connection is restored and the transaction is relayed out resulting in blocks that have confirmations for those transactions.
Because the blockchain can be transmitted on a thumb drive, you could cut the time down before connectivity is reestablished by having the next motorcycle delivery bring a copy of the blockchain from somewhere that had connectivity. The motorcycle could then also take a copy of the desert island village's blockchain back to where connectivity exists so those transactions will get confirmations in the longer blockchain.
Spend transactions could even be sent to the main network via smoke signal if you needed as there's not a whole lot of data. A text message holds 160 characters and a raw Bitcoin transaction is typically under 500 characters. These can be created using BrainWallet.org:
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http://brainwallet.org/#txBut overall, bitcoin is an online digital currency and doesn't work for desert islands without at least sporadic (e.g., hourly) occurrances of connectivity. Casascius physical bitcoins (or similar) do, however, work in such an environment.