What is going to happen when it is totally infeasible to download the entire bitcoin transaction history? As far as I know, the bitcoin system currently relies on the principle that all transactions are made by someone with access to the entire history of previous transactions. Since bitcoin's public debut 2.5ish years ago, these records have stayed small enough that it's still easy for one person to download and update it.

Once it gets large enough, though, it's going to be hard to do that. When I ask most people about this problem they just tell me that eventually we will start trusting our money to some bank-organization-type-thing (e.g. Coinbase) and we won't have to worry about the integrity of our records.

Imagine that in the distant future, the bitcoin system becomes widespread enough that the rate at which transactions history grows exceeds the rate of change in disk space technology. (far cry, I know, but let's just assume a worst-case scenario.) How would bitcoin account for this problem?

Is a better, more widely agreed on solution?
The default answer to scaling is SPV (lightweight) clients, which allow very good security and control of funds without the entire blockchain. Full nodes will still exist but the cost of running one will not be prohibitive.
In practice many people will use some sort of bank service. If the funds are in full control of the bank it has some disadvantages, but yes - in this case of course the banks will settle intra- and inter-bank transfers with internal records and reciprocity agreements, not with the blockchain.
There are some schemes which allow making a payment without it being recorded as a transaction in the blockchain, and without having to trust a bank with your money. This will greatly reduce the total size of the blockchain.