A dictionary is not the appropriate data structure for the operations you are describing.
If the intervals are required to never overlap then you can just build a sorted list of intervals and binary search it.
If the intervals can overlap then you have a more difficult problem to solve. To solve that problem efficiently you’ll want to build an interval tree:
http://en.wikipedia.org/wiki/Interval_tree
This is a well-known data structure. See “Introduction To Algorithms” or any other decent undergraduate text on data structures.