Oh dear - poor girl got more than she bargained for!
- but I'll repeat my ways of hand tapping M4 holes, which work for me.
For softer stuff like plastics and softer ali I use a 1/8th inch drill (which is probably less than the minor diameter), unless it is brittle plastic, when I'd use 3.3 or 3.4 mm.
For normal materials like mild steel and hard ali, I use a 3.3 mm drill.
For hard stuff like 300 series stainless, inconels and the like, unless it needs to exceptionally strong, I might use a 3.5 mm drill for rough work. If it needs to be strong, I'd use 3.3 mm and take extra care. A set of serial taps, which come in sets of three so the smaller two are undersize can help here, as can thread forming taps if the material is sufficiently ductile.
I'd never use a 3.5 mm drill for ali (unless I didn't have one the right size). 3.3mm or 1/8", depending on the hardness and the depth of the tapping - I'd go for 1/8" (and probably a forming tap) in ali up to 1/8 in thick.
Most times when I've broken a tap, which is infrequent these days, the reason is either that it has gone in askew or that the force applied is not in line/at right angles.
If at all possible I tap in the lathe, bench drill or mill in order to get the angle straight, and turn the tap by hand, to be able to feel the force used.
If I have to tap in the field, so to speak, I like to use HSS taps because they are stronger than carbon taps and take slight sideways forces better.
If it's getting extra hard to turn there is often a reason not related to the size of the hole. Withdraw the tap, have a look at the setup, clean off the chips from the tap and the hole, lubricate and try again. Never force it - it's a real hassle when you break a tap.
Lubricate always, except for some plastics.
-- Peter Fairbrother