What happens if two miners mine the next block at the same time?
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).