This dissertation explores fundamental limits in radio astronomy and develops techniques that utilize the scintillation of compact objects to probe detailed properties of their emission regions and of the scattering material. I develop a statistical framework for observations with spectral resolution at or near the Nyquist limit, suitable for describing the observed statistics of strongly-scattered sources. I demonstrate that these statistics can effectively isolate the signature of an extended emission region, requiring no assumptions about the nature or distribution of the scattering material. Then, using observations of the Vela pulsar at 760 MHz with the Green Bank Telescope, I thereby achieve a spatial resolution of 4 km at the pulsar. Finally, I explore the signature of refractive scintillation on the interferometric visibility measured on long baselines, and I derive optimal correlation estimators for quantized data.