What would cause the bottom half of a radiator cool down after heating up?
One pipe steam pipes have to maintain a consistent pitch down from the radiator to the boiler. My guess is when you put the new radiator in you let that existing pipe drop down a little bit. Now it starts to get hot but once the little well of condensed steam(water) forms at the low point, it gets blocked and stops getting hot. 

Another possibility is that the vent is closing too soon and not letting all the air out. Should be an adjustment.