If you were to run out of space, the host will most likely give your refund but it's your responsibility to make sure you pick a server which meets your sites requirements. If you have enough space, but the server says that it has no more space then the refund should be offered, or you get your server upgraded for free.
as @Malcolmjr96 mentioned refunds are usually offered under these circumstances, so either upgrade if you are happy and satisfied request a refund and transfer. (most hosting providers offer free migration)