As long as there is demand for Bitcoin transactions and the hash rate is somewhat stable, Bitcoin will confirm 1MB of transactions every ten minutes in average. Since this is a fixed supply, Bitcoin would not be able to satisfy a demand greater than 1MB per ten minutes and some transactions would never confirm. This is a graceful failure, as it would be rational for miners to confirm the transactions that give them the highest revenue, and thus the transactions that are the most valuable to their users will be catered to first, and transactions that signal less priority get priced out of the blockchain first. These would either move to off-chain solutions or other blockchains in the long term.
It seems obvious that fees would thus adjust respective to blockspace demand and economic value of the transactions. As either grows, pressure to provide more capacity will mount. It seems likely that sufficient pressure would lead to adoption of some capacity increase eventually, be that Segregated Witness an Lightning Network, or increased demand for altcoin solutions.