Presumably quantum mechanics really is the way the universe works, and it appears we don't really understand where the boundary between quantum mechanical phenomena like interference end and classical phenomena begin, but presumably it is not a discrete transition but rather a smooth transition from quantum to classical. Therefore, shouldn't there be computers which are intermediate between "fully quantum" and "fully classical" devices? and might have some intermediate speed-up over fully classical?
You're presumably thinking of a spectrum with classical mechanics at one end and quantum mechanics at another, with some hazy "classical-quantum" in between. That's not a great way to think about it. Classical mechanics is more of a practical approximation of quantum mechanics under certain conditions (cf. classical limit), as per the correspondence principle. Even in semiclassical physics, one part of the system is approximated to be classical (say external electromagnetic fields), as it's a pretty good working approximation anyway, while for the other part the precision of quantum mechanics is required (e.g. electrons in an electromagnetic field).
There certainly are "semiclassical methods" of computation that provide speedup over regular classical computation but are, in theory, slower than full-fledged quantum computation. For instance, the Fourier sampling in Shor's algorithm for factorization on a quantum computer can be carried out in a semiclassical way by using the “classical” (macroscopic) signal resulting from measuring one bit to determine the type of measurement carried out on the next bit, and so forth (cf. Phys. Rev. Lett. 76, 3228). However, this is quite different from your mental image of computers working in some transition regime between classical and quantum physics. In essence, it's just that for some part of the computation we're using classical techniques while for others we're using the speedup of quantum algorithms.
All quantum computers now are partly classical. The experiment is set up using classical math libraries like Numpy.
I think your instinctive "ur-question", i.e., "is there really a clear line?", is a good one.
As time goes by (as the song goes) the line will no doubt get very blurry. Or perhaps better put, our understanding of concepts like "classical" and "quantum" will be transformed by our interaction with these devices.