Capacitance depends on diffusion of charge carriers, so has a lossy diffusion characteristic at quite high frequencies (100s MHz?), regardless of doping. (Purpose made varactors are low voltage devices, so have stronger doping and thinner junctions, and are good to much higher frequencies. They're probably also constructed differently, having lower series resistance.) At modest frequencies, and low enough amplitudes, it truly does look like a pure capacitance.
Typical recovery for a 1N400x series is several microseconds, so one wouldn't be able to be tested with a pulse repetition rate much over 200kHz.
Tim