You can even send transactions with no fee at all, but they tend to be prioritized rather low. So far, every transaction get processed. Eventually. I've seen

You can even send transactions with no fee at all, but they tend to be prioritized rather low. So far, every transaction get processed. Eventually. I've seen some no-fee transactions take several hours recently. But there's also currently some issues with the overall load of transactions, due to a gambling site spamming tiny "dust" transactions as part of its system. That's caused there to be a lot more transactions, which means sometimes you'll have to wait a number of blocks for a no-fee transaction to process. A block is supposed to happen an average of every 10 minutes, but it's not deterministic and there's a variance. It can take an hour or more for the next block to come up, in rare cases.