You're right, I mixed two concepts here. The client software is designed right now with a lot of flood-defense features, because it is known that the network is not hosting the volume of transactions you're talking about. ATM, it is assumed that if there is that much traffic, it is someone trying to spam/flood the network, not legitimate users. However, if volume legitimately increases, then surely the devs will loosen up a bit on those defenses...
I'm not entirely up to date on leading theories for dealing with massive transaction volume (i.e. if BTC were trying to replace all CC transactions), but much of it has to do with converting the standard user to lite nodes (not storing the entire blockchain, only scanning and keeping what is relevant), and only the miners/pools need to store the whole chain.
Two ways that I'm familiar with are not particularly easy: the blockchain could be converted into distributed storage, so that very few nodes will have to maintain the entire thing. Alternatively, the way transactions are executed, allows for pruning of old transactions. Individual nodes could do this without problem. But if tx volume really gets big, even expensive nodes might not be able to handle it, and it might be necessary to reach some kind of global agreement to forget the distant past, and "lock in" the current state of the network at a given time. This would be a major coordinated effort/problem, but if it is "solved" this could allow the network to reset the storage requirements every couple of years...
I'm sure there's other options and theories, but I haven't been following it too closely. You might consider starting
here, where you can see a well-known security researcher asking the same question. I'm sure there's a lot of discussion on this in the forums.