I never over run my jobs. On my contract, it specifies that there is a guarantee maximum price unless there are changes, and there always are, and hidden issues the none of us can see. If I exceed the Guarantee Max Price, I pay out of my pocket to complete the job. I have to do that only one time.
How big is the house? More info is needed to get remotely close to getting a ball park number. Since its NYC, getting materials and parking maybe an issue and it should factored into the estimate. Planning with your contractor will be the best thing you can do. Then do it some more. On a job like this, request a rough time line and try and stick to it as close as possible. Lots of residential contractors make their money by taking on too many jobs but have only a small crew and rely on sub contractors. Subs have other work and that's where things go bad. Wait time cost money, sometime lots of it.
Higher end materials on a kitchen normally run about 70-80K for a typical 3/2 house. Appliances will eat up 20K easy. Baths are 20-50K and it only goes up from there.
My advice is to have some kind of plan to get rough estimates if you do not already have a contractor. From my experience, big contractors or companies "usually" do not provide enough details on higher end remod jobs. Who's doing the design work? Who's providing the details? the more specific, the less problems unless the contractor's is a savvy designer to catch design mistakes if there isn't a designer involved.
I really don't think its a 400k job you have there. But then again, we need a lot more info and the size of the place. Wait, 400k? I'll fly out and do that for you right now
. I hope this will help you with your questions.