Overhead lights don't work but voltage meter shows sockets have power
Voltage testers can sometimes be tricky because they can show "phantom voltage" induced (well, capacitively coupled) from other wires.  

However, the more likely problem is -- remember, current flows in loops -- and the safety ground wire is not a player.  So somewhere the loop is broken; if you're measuring voltage (hot to ground) then it's likely that side of the loop is OK. Check the neutral side. 

The neutral side is trickier to test, because it's supposed to be near 0 volts.  However if it is broken, and a load is connected and turned on, then a disconnected neutral will float near 120V, and will give a voltage reading.