The answer to your question is no, that is not normal or possible with a board that is well designed and well-made. The plated through hole would not be permeable and there would be no possibility of solder flowing through to a plane layer. It is possible for this to happen if the board is not well made or not well designed with adequate clearance around plated through-holes. If the plated through hole really did short to the plane, then you can try drilling out the plated through hole with a small drill. If that makes the short go away, then it was most likely a plated through-hole problem.
In general, if you want to get to the bottom of this, you will have to be systematic about it. If you haven't already, do a careful visual inspection of the whole board looking for solder blobs.
A careful inspection of the board layout might be a good idea also, to look for places where power and ground are, perhaps, closer than they should be.
If there are other components on the board, one of them could be responsible for the short. Removing all the components might be a good idea. Check the short after each removal. If the short goes away as you remove components, then it could be a solder bridge hidden under a component or a bad component. Or maybe a backwards diode.
I don't know what equipment you have, but you could also try applying a high current to the short and use a thermal imaging camera to find a hot-spot on the board. Most likely the short area will appear as a hot spot because the current density will be much higher in that area. It is a current bottleneck. It might only be a few degrees hotter than the rest of the board, or it might actually get quite hot.
If the short is definitely in the board, and the board is large, you can repeatedly cut the board until you are left with only a small segment that has the short in it. Then this area can be examined very carefully, and hopefully the short can be discovered.
Hopefully one or more of these ideas will help you.