Sure, a base die rank can be lower than a specialization die rank. Take a real world example to illustrate the idea. base: driving wheeled vehicles> driving automobiles > driving Ford Taurus. You can see there that driving a semi rig is included in the general category, but without some practice you would not drive it as well as your car.
Right, that's why you have extra dice in driving your car.
You could have Driving Wheeled Vehicles 1D12 > Driving Autos 1D6 > Driving Ford Taurus 1D4. You're only rolling 1D12 to drive a semi, but you're rolling 1D12+1D6+1D4 when driving your Taurus. This sort of dice chain represents a specialization on top of a very good general ability.
But you can also be specialized in a specific area without having much general ability. The driving example isn't the best way to represent this sort of dice chain, so let's use Knowledge instead. Take the ADC Knowledge 1D4 > History 1D8 > WWII 1D12. This person knows a lot about World War II specifically, a fair amount about history in general, and very little about fashion trends among modern Japanese teenagers, for example, or anything else not directly history related.
Or for a more combat oriented example, let's say Melee 1D6 > Swords 1D4 > Fencing weapons 1D12. This is a fencer who has spent hours upon hours fencing, and had little other experience with other weapons. Obviously, the fencing weapons specialization is the highest. To some extent knowledge with a specific weapon will carry over into combat in general, hence the 1D6 basic melee. A fencer has probably at some point in his life picked up a few other swords, or at least paid attention while watching other people use them, so Swords of only 1D4. A master fencer could very well be a total klutz when he picks up an axe.