A lot will change in the crypto space before the last Bitcoin is mined. The revolution is DLT (Distributed Ledger Technology), not blockchain. Blockchain is merely the form of DLT that Bitcoin uses. The drawback to blockchain is that the network must wait 10 minutes before adding each consecutive block. This is to ensure that multiple computers do not add the same transaction to the blockchain at the same time. This is the primary purpose of the minors. This is also why the mining difficulty is always increasing. The network must always have a ten minute pause between each block.
If the miners only receive transaction fees for validating transactions then the people using Bitcoin will ultimately decide how much money they are willing to spend on transaction fees, verses using something else. Many miners will likely quit, thus driving down the cost of mining to the point where those who remain will profit.
So, to answer your question, because the network will always need a 10-minute pause between blocks, the only way Bitcoin will fail from miners quitting is if the transaction fees become so low you can't get enough miner to prevent a 51% attack.
There is a similar question which is a community wiki.
– Shelby Moore III – 2013-03-27T21:34:20.373Related: http://bitcoin.stackexchange.com/questions/5275/what-will-happen-to-mining-after-the-20-999-999th-bitcoin
– Highly Irregular – 2013-03-28T02:37:14.980