I understand that the time resolution of a spectrogram with no overlap is the sampling frequency divided by the window length. How does this change with, say, 50% overlap? Does this double the time resolution? If so, why can't I just have a large overlap and window length to achieve both high time and frequency resolution?
A sliding FFT (maxed-out overlap of N-1) used for a spectrogram will turn a single sample event into a fat wide hump (shaped like the window applied) of width 2N-1. You might call the fat wife blurry peak of a single hump “resolution”, but multiple overlapped events of different heights plus a bit of noise might make it impossible to “resolve” (as in separate) adjacent event peaks in time.
To make the blurry humps for short events appear narrower you have to use shorter FFT windows, no matter what the overlap.
A 50% overlap does provide more time resolution, as the typical quantized Von Hann window eats edge events. The extra windows in a 50% overlap brings that info back up into the picture, as the sum of 50% overlapped Hamming or Von Hann windows has a constant gain.