I expected unbreaking three to multiply the length the armor lasted by 4.

What actually happened was the lifetime of the item was extended, but not by a significant margin. (less than 50%)

I tested this by having a piece of armor with unbreaking 3 and a piece without unbreaking 3, and fighting mobs. The unbreaking 3 armor(chest) also had protection 4, while the other piece had protection 3(legs).

The Chest experienced 46 durability damage.
The Legs experienced 63 durability damage.

Further Testing(20 hits taken):

Chest Starting Dur.:236 Final Dur.: 223 Dur. Lost: 13
Legs Starting Dur.:186 Final Dur.: 166 Lost: 20

Continued Testing(200 hits taken):

No Unbreaking: 200 damage(feet) and 200 damage(legs)
Unbreaking 3: 124 damage(helm) and 137 damage(chest)

Continued Testing(227 hits taken)

No Unbreaking: 227 damage(feet, legs, helmet)
Unbreaking 2: 166 damage(helmet)