In modern football, what is an acceptable length of time to let a manager prove himself?

It depends really. Nigel Clough had 3-4 year. The board didn't even care how he did in his first and second full season because it was a long term project but there was no improvements whatsoever, they actually gave him 3 seasons to turn it around but they were still the same old side, with but more young players. Some thought it was harsh, others thought it was the right decision. Same as Mowbray for Middlesbrough. It also depends on the target right. I mean look at Jol, some said right decision, others say it was harsh because he is a good manager. I thought it was harsh only because the club lack ambition, why else would Hughes leave.

For a top club like Man utd, I say one season.

For a top club like Liverpool who were realistically after top four first, two seasons.

Surviving relegation, now that is the hardest. Me personally, I rather they don't sack the manager, because they deserve to be there since they help get the club promoted.