Want to know the most common planning mistake? It isn't the budget. It's trying to squeeze the northern circuit into four days.
Under six days
It's possible, but it turns into a driving trip. You can do Tarangire and Ngorongoro properly in four or five days from Arusha and have a really good time. What you can't do in that window is add the Serengeti without flying, and flying changes the budget.
Six to seven days
The honest minimum for the classic circuit: Tarangire, Ngorongoro and the Serengeti, with two nights in the Serengeti instead of one. With only one night there, you'd arrive in the afternoon and leave after breakfast, and that's a long way to drive for a single game drive.
Eight to ten days
This is where the trip stops feeling like a schedule. With three nights in the Serengeti, you can spend a whole day in one area without worrying about what you're missing. And that's when the best sightings tend to happen: you find something in the morning, and you're still with it in the afternoon.
Twelve days and beyond
Now you can add something really different: the southern circuit, a walking day, or Zanzibar at the end. Adding a fourth northern park instead is usually the weaker choice. Once you've seen the Serengeti properly, each extra park adds less and less.
Count your days honestly
Don't forget the flying days. A “ten-day safari” that includes the flight from Europe at one end and the flight home at the other is really eight days on the ground, and you'll probably sleep through the first one!