It's possible to configure a timer in input capture mode to measure the frequency of the encoder pulses, which allows determination of speed. Of course, if the acceleration isn't zero, then you get measurement errors. Some kind of algorithm may predict the acceleration and compensate the speed measurement errors. The success depends on the fitness of your algorithm to the dynamics of your system.
In addition to frequency measurement, some other logic must also determine the direction of rotation. Maybe another timer in encoder mode, or some external logic circuit can be used.
Position measurement is even harder. Between the pulses (call it the silent time), you also need a good prediction for speed. This may be relatively easy when you have constant speed, but gets more and more complicated if your speed & acceleration aren't constant.
If your encoder isn't continuously rotating (if the speed becomes zero), I can't imagine a way to over-sample the position, as you are always in the silent time. If you have a good mathematical model of your system, maybe you can make some predictions.
But after some point, the problem becomes a system dynamics & control theory problem. As your accuracy depends of the performance of your prediction algorithms, I doubt that you can rely on that additional resolution you think you will achieve. If there is too much uncertainty in those predicted/over-sampled bits, it means you don't actually have them.