They aren't mutually exclusive and I'll give you 3 examples of varying difficulty.

People fail the DPS checks in say, A3s, because they can't actually handle DPSing while doing mechanics. They either tunnel vision or fail mechanics, leading to deaths, DPS loss, wipes. I've never heard of a group disbanding there because they couldn't make the enrage timer. When you get there you'll eventually win.

People fail the Sable Price in the Aery all the time, not because the check is particularly hard but because they have to start paying attention to what is happening around them while doing their rotation half-correctly.

Another DPS check that I see people fail so much that I have to issue a warning whenever I get it is Ramuh's Grey Arbiters. People never fail it because they lack the DPS to make it, it's extremely low. They fail it because they can't be bothered doing the mechanic, which requires peeling off Ramuh and killing the Arbiters.

DPS checks usually aren't the problem.