What happens if two miners mine the next block at the same time? The process of mining as described in the answer to this question is simultaneously repeated by multiple miners.

Is it possible for two miners to find the next block at the same instant? If it is, what happens if they do?
Yes, it is possible, and you can actually follow "orphaned blocks" here: https://blockchain.info/orphaned-blocks

Bitcoin clients always trust the longest chain, so if two blocks is mined on the same time, it's up to (51% of) the miners to decide which is going to be 'accepted' and which is going to be worthless.

This is one of the reasons why you shouldn't trust only a few confirms, as the confirms could be a orphaned blockchain. After "enough" confirms, everyone will mine on the "correct" blockchain, because any work(mining) on a orphaned blockchain is waste of power (unless your trying to perform a 51% attack).