According to the spec sheet here at Intel, the TDP on those CPUs is 150W each. It quite possibly is a cooling issue and may require better air cooling or even AIO liquid cooling. Also look at this forum post. There were 2 heatsinks available for the T5600. One for lower end CPUs and one for higher end such as the CPUs you installed. The high end heatsink part number is 1TD00.
I upgrade a T5600 with 2 E5-2687W cpu and when I booted the computer up it started running very slowly and never got past the windows loading window when I touched the computer case it was quite hot. I quickly turned it off.  I have the upgraded psu 840W as I saw that there was a thermal issue with the lower wattage psu.  I’m pretty sure I applied the thermal paste as instructed, (although it was my first time doing this). Any ideas on where to start to trouble shoot this?  
T5600 too hot after cpu upgrade