It sounds like you may want to make what's called a "bottoming tapped hole." This is blind hole (that is, a hole which does not extend through the material and out the far side--so you can't see through it, therefore "blind") where there is a maximum length of thread, in a minimum length of hole. To do it, you need a regular tap drill, and three taps-- a taper tap, a plug tap, and a bottoming tap.
A taper tap is the one you normally buy in a hardware store. The other two are bought from an industrial supply, or could be made from an old broken taper tap if you have a grinder.
A taper tap usually has a cone portion at the tip. This is followed by the threaded portion. The beginning of the threaded portion, though, has a chamfer ground on it, so that it increases from a small diameter to full thread diameter. It is this portion that does the thread cutting, and is also the portion that results in "incomplete threads," since the material does not end up with the threads cut to full depth.
On a taper tap, this region extends across the first 7-10 threads at the tip, so it leaves behind 7-10 incomplete threads (threads not fully cut to full thread profile/depth). Because of this, when using a taper tap only, you've got to tap drill far deeper than the intended final thread length.
A plug tap and a bottoming tap both have little or no cone at the tip-- they get right to the threads. Plus, the chamfers are cut steeper. The plug tap has only 3-5 incomplete threads, and the bottoming tap has 1-2 incomplete threads. They each have less ability to cut threads than the previous one, so you have to use them in order.
So, to make a hole that is threaded nearly to the bottom, you
- drill just a little longer than the intended thread length
- run down the taper tap
- run down the plug tap, bringing the fully tapped region deeper
- run down the bottoming tap, bringing the fully tapped region very close to the bottom.
Along the way, you have to be even more careful about chip removal than you would have to be with a simple taper tap in a blind hole.