Bridge rectifier outputting 338 volts DC with 120 volts AC input
Sounds like you have something resembling a voltage doubler, albeit one with very little current output capability. 

The top schematic is a conventional voltage doubler as found in classic 120/240 power supply designs with a voltage switch. 

The bottom schematic is what  you have described. 







simulate this circuit &ndash; Schematic created using CircuitLab

Here is a simulation of the output voltage of each circuit: 



The blue trace represents the top circuit, and it provides a smooth DC output if the capacitor is large enough in relation to the load. The orange trace represents your described circuit, and as you can see, it averages out to almost the same voltage. 

If you put a substantial load on the bridge output the voltage will drop to closer to the average (not peak) abs voltage on the mains, more like 108 VDC.