In the following I will assume some basic understanding of how Bitcoin works. If you feel like you are missing some knowledge, watching this video will likely be the most productive 22 minutes of your day. If you feel like you have knowledge that is superior to mine, please feel free to correct any mistakes, pop some FUD and hopefully tell me why I am wrong.
Because I would really like to be wrong about this.
Every 210 000 blocks, that is roughly every four years, the Bitcoin block reward is cut in half. If the Bitcoin value doesn't doesn't increase exponentially until forever and after, this means that the amount of electricity and hardware that can be paid for from the reward of a block will drop until it becomes effectively zero. The date when this happens depends on various factors, but it is probably safe to say that most of us will still see it happen in our lifetime.
Hardware investments and electricity spending translate into hashing power. The amount of money spent on mining operations is directly related to the integrity of the network. Once a single malicious entity is able to mine with about 50% of the hashing power, this person can have its way with the block chain.
For Bitcoin to be safe, the monetary investment to create and uphold ~50% of the hashing power must be impossible for a single malicious entity, even when the hardware is slowly being refinanced by mining rewards.
Bringing these two things together, it becomes obvious that a significant amount of money (= electricity and hardware) needs to be spent each day in order to secure the network. Since the block reward system is coming to an end, transaction fees are projected to replace its function. If a sufficient share of the value of each transaction is given to the miner that includes the transaction into his block, this will create a mining economy big enough to secure the network.
In reality, the initiator of a transaction is free to include whatever fee he deems appropriate. And the block creator can choose to ignore transactions with low fees. At a first glance, this might seem like a good solution. But we are dealing with a textbook case of the prisoners dilemma: While it might be in the best interest of all miners to only include transactions with high fees in order to drive prices up, this will not happen because every miner by himself profits most by also including cheap transactions.
As a result, mining revenue will drop first, then the hash rate and the difficulty. It is true that with sinking rewards the refinancing time for a 50% hash power mining pool would still remain unchanged, since this is an equilibrium that is created in relation to other investment options. But at some point, it simply doesn't matter. If the requirements drop low enough, somebody will be able to shell out the millions to obtain 50% mining power in a single strike. This will be the beginning of the end of the currency.
PS: A limitation in block size (as it is currently implemented in the standard Bitcoin client) could act as a mechanism to create nonzero marginal costs for including a transaction and thus increase the transaction fees. However, constantly hitting the block size limit might lead to the opposite problem of making small transactions unaffordable. Do we want that? (Maybe). It would also require some form of central regulation since there is no such block size limit included in the protocol, let alone a mechanism to adjust it to a changing demand.