Could Bitcoins (or any other blockchain based cryptocurrency) feasibly ever be used for purchases that need to go through fast? Today, one of the problems with cryptocurrencies like Bitcoin is that it takes quite some time for a transfer to go through and enough confirmations to take place. This really isn't a problem when it comes to inherently slow buying procedures, such as for example ordering something online and receiving it a couple of days later by mail. However, other types of purchases requires a much faster process (for example if you're buying a digital good and want it immediately or if you're buying a physical good and want to leave the store with it right away), and as it is now, cryptocurrencies don't seem suitable for these as it simply takes to long time to confirm the payment and defend against double-spending.

Is this a problem that is inherent with the Blockchain technology, or could purchases go through much, much faster (at least so that they can compete with credit card transfers) given enough resources or faster computers in the future?
Blockchain technology, at least without accepting central party that controls access to the chain, inherently has slow blocks, or strong centralization incentives.

The reason is that blocks need to propagate much faster than the block interval time. If they don't, miners who are further away from the majority of the hash power are put at a disadvantage.

To explain this intuitively, notice that when miner A creates a block, he can start working immediately on a successor, while another miner B first needs to wait until A's block reaches him. During that time, B works on an outdated chain, and is unlikely to produce a block that ends up being accepted. The closer A and B are located together, and the more trust they have in eachother about not lying about new blocks they produce, the more they start to behave like a single party.

Thus, the ratio of block interval and propagation speed across the network is a means for keeping the advantages of larger miner groups over smaller ones in check.

This does not mean that the bitcoin currency is inappropriate for fast payments. They just require another mechanism than on-blockchain transactions. For example, look at the proposed Lightning network, which allows many small transaction between parties to occur, while only settling the resulting balance on the blockchain. It requires more trust, but has higher performance and convenience, and lower costs.