When a board takes the decision to sack a manager, it's merely the start of a process. The next step -- finding a replacement -- is the trickier part of the job and often determines whether the initial sacking was the correct decision.
Premier League chairman might consider the story of Wolves in 2011-12. They dismissed Mick McCarthy in February, announced they were looking for an experienced replacement and then realised they couldn't lure anyone appropriate.
Eventually they appointed McCarthy's assistant Terry Connor, who didn't even want the job. Results got worse and they went down. If they couldn't find anyone better, they should have just kept McCarthy.