If your characters are settling down for a long rest, but are interrupted four hours in, it can be a question of it they get the benefits of a short rest. Sadly, according to the rules, a short rest and a long rest are two different things. Even if a long rest is interrupted, the party doesn’t gain the benefits of a full short rest. Of course, there’s nothing stopping a DM from changing the rules or throwing them out entirely, but that is how the rest mechanic works as written in the game.