I'm coming at this from a point of view that assumes Bitcoin has been widely adopted as a currency worldwide and is essentially mainstream. Every day people are using Bitcoin in the same way that they use their credit cards today.
Will the transaction volume at this level be sufficient to overwhelm the Bitcoin network given that transactions are processed differently to those of the credit card companies?
There are three parts to this:
The first two are pretty simple to solve since most computers already can process up to 100 transactions a second (resulting in roughly 10*60*100=60'000 transactions per block) and this can be pushed by investing in larger hardware to do the processing, and offloading verification to trusted third parties.
The last part is harder to solve as it results in possibly O(n^2) messages going around the network. Here a more structured network might do the trick, but all attempts so far to create a structured network have been dismissed.
Last but not least the blockchain grows linearly with every transaction which uses diskspace on every client. This has been partly solved by Satoshi in his paper.
There are actually a number of possible threat models to evaluate. But since you asked about overwhelming "the network", I'll look only at attempts to overwhelm the network as a whole and not talk about attempts to overwhelm a few nodes. (You can overwhelm a node with invalid transactions, but that won't hurt the network as a whole.)
First, nodes will only pass verified, valid transactions to other nodes. And a node will only pass one transaction that attempts to claim a given output. And further, a node will not pass a transaction that claims a new output (one that hasn't been stable for a long time) unless it includes a transaction fee.
You have to assume an attacker will use no transaction fees. Otherwise, he'll go broke pretty soon. So in order to get his transactions relayed, he'll have to use old outputs. The problem is, his transactions will get into mined blocks, consuming his old outputs and replacing them with new ones. With new ones, he won't be able to get his transactions relayed for free.
The net effect is that you can't sustain such an attack for long because you can't have that many old transaction outputs that you can claim, and you'll burn them very quickly because you can only relay one transaction per output.
Effectively, after a short initial burst, at best you can send new transactions as fast as they're included in mined blocks, which might be maybe 100 transactions every 10 minutes. That's not much load.
The answer is no, and here is why: average computer can verify about 80 transactions per second. But most miners now days connected to pool servers, and in that case transaction verification happens only on the server, which could have a very sophisticated hardware and process much larger number of transactions.
