I would. And if you have some RAM you know is good, or if you can test the RAM you're using in another computer to see if it works that's even better.
It's no fun to remove the cooler, and you introduce a new possible problem of the CPU becoming seated improperly, but you need to test it. You have 4 sticks of RAM and 4 slots they're in, that's 8 possible points of failure. I would try 1 stick of known good RAM in each of the 4 slots with no hard drives, gpu's, etc. connected. Just absolute bare-bones system with the minimum possible requirements to POST and get into your BIOS.