The target comes from the manager too, though. After finishing 4th last year and with reinforcements over the summer, Conte's message could have been "we'll give it our best shot. We don't have all the pieces in place, but with a full pre-season and reinforcements I expect us to do better than last year and, if luck is a bit on our side, we may achieve silverware too." Instead he was downplaying our chances the entire time and talking about how top 4 is a trophy. Yeah, it was the previous year given what he inherited, but this year he should have set his sights higher. But he didn't. Was it to cover his ass? Was it because he didn't believe it himself? I don't know. But it's definitely not because the club went behind his back and told the players "don't worry lads, all we need is top 4."
In most places that I work our bosses set stretch goals to motivate us to do better. That is the manager's responsibility in a football club. But if you're afraid to stick your neck out lest you tarnish your reputation, you'll downplay the goals so that you end up looking good in the end. And that's squarely on Conte.