I have also posted here as you have two threads on same subject.
You have to think about wheels and tires as a whole diameter; just because you upgrade to a 16" rim doesn't mean the wheel as a whole gets bigger. Your 15" rim with 225/75R15 right now is 28.29" in diameter. If you upgraded to a 16" rim but mounted a 225/70R16 the wheel only grows to 28.40", an increase of only .11 or 11/100th of an inch. Any wheel/rim app will help you understand this concept.
http://www.wheel-size.com/calc/
As to why you have so little space? Are you sure the sidewall of the tire isn't behind the wheel trim?
Cheers
Tony