Why does solder not conduct sometimes?
Sounds like a classic cold solder joint. This happens when the solder and pad are insufficiently heated (time or temp), or the surface is not clean, and the wetting action does not occur properly.

If you use a meter to measure from one point to another on the solder itself, it should conduct, but between the solder and pad, there is not actually a good electrical connection, hence the open.

It's not that solder isn't conducting, but that it hasn't made a bond to the copper and therefore contaminants or flux are actually creating a non-conductive barrier.

As for the pin-to-pad conducting but not solder-to-pad, the pin is resting on top of the pad and likely making a physical connection (held in place by a brittle solder blob). The solder, however, may have a thin air gap left over from flux or contaminants, and isn't actually making contact with either.