AC ripple at input of linear regulator
Look at the datasheet.  Every decent linear regulator datasheet will specify the headroom, or minimum input voltage required for the regulator to produce the specified output voltage.  In your case, the dips of the voltage between when the input cap was charged went too low and the regulator couldn't hold up the output anymore.  The larger cap raised the level of the dips to above what the regulator needed.

AC on the input of a regulator even when this is above the headroom limit still matters, but there is a different spec for that.  No regulator is perfect, so some amount of the input AC component will appear on the output.  Sometimes this is specified as a input rejection ratio, or sometimes as a "input regulation" spec.  For example, if the input rejection ratio is 60 dB and the input voltage varies over 1 V, then the output can vary by up to 1 mV from the variation of the input voltage alone.