James: You didn't mention the year or model number of your truck. Some trucks came with (or had dealer installed) HD radiators that had the radiator pressure valve shown on the left side of the top tank in your picture. In that case the radiator cap would have a gasket that would not allow overflow at the radiator fill tube. The petcock on the right side of the top tank is to check coolant level. Listing below is for the pressure relief valve. GMC Maintenance Manual says the valve is set for 4 psi. There are a few paragraphs about the pressure valve in the Maintenance Manual, but it isn't rocket science. Let me know if you'd like me to scan and post them.
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'57 GMC 102, Original 347 V8, HydraMatic, 3.08 rear gear, added A/C, disk front brakes, HEI, AFB carb, '98 Honda Black Currant paint. T-boned and totaled 10/12
'52 GMC 152 Stake Bed, Original 228, SM420, added A/C, HEI, disk front brakes, '67 Chev 3.55 rear gear. Gets used as a real truck.