It is known that the Grey squirrel population is out of control,&that even with predation from the top raptors makes very little difference to their numbers, but I think other factors have to be considered, just because the birds nest & lay eggs doesn't mean that the fertility is good or they will have a good hatching, many things can cause this, weather, food quality, age, stress, habitat predation etc.
Some species of birds take years to recover from a bad winter,& although we love the heat in summer for some birds it's very often a killer for their food source at a crucial time for feeding themselves & their young, coupled with that they then cannot build up their fat stores to help them through the winter. In our constant quest to have a weed free spotless garden then that also affects the butterfly & other insect population, the same applies in the countryside with crops the ever increasing push to get a crop harvested & another one in doesn't give the wildlife a chance to complete it's life cycle in time. The Harvest mouse is a prime example of this practise. The food chain is a very fine balance destroying one species can then lead to others being destroyed.