Translation:He will not stay longer than four days.
In this case it sounds as if the 'nicht bleiben' is a unit, as if the 'nicht' only negates the 'bleiben'. In the English sentence 'not' is negating the whole sentence. I hope you get what I mean, it's difficult to describe. :)
Take the difference in the meaning of those two examples:
Er wird nicht drei Tage (lang) essen. (He won't be eating for three days (it'll take him only an hour to eat)
Er wird drei Tage nicht essen. (He won't eat in the time of three days.)
Difficult to explain it is indeed. :D