I recently went to start up my 1967 mustang coupe after about three months of sitting in the garage, but when I turned the key nothing happened, there was absolutely no noise. I hooked up the charger assuming the battery was dead and the charger immediately Indicated that it had a full charge, so I tried to start it again with the charger hooked up and this time got a weird buzzing/clicking noise coming from the solenoid, but again it didn't start. I replaced the solenoid and battery cable brackets that connect to the battery terminals and still had the same problem. The battery is about two years old, and when I checked it with a multimeter it read 11.9, so I'm not sure if it is the battery or something else.
Its possible that your battery has a dead cell which unloaded it could still give you a high voltage reading but as soon as you apply load it drops drastically. Put your voltmeter on your battery and watch it when your try to start it. If it drops way off say down to 9 or 10 volts or less then you may have a dead cell.
Another thing that could be going on and is that as cables get older they corrode within the insulation. If you have old battery cables they will corrode over time and then they will not carry the current they need. You can't always see this corrosion but sometimes you can see this between the strands of wire either a white powdery buildup or even a greenish buildup. Check all the cables even the one going to your starter.
I would also check your connection at your starter and grounding point to your block to make sure they are tight.