The following information is based on my experiences, and does not have a ton of scientific study information.
The following information is from many different loads, barrels and frames in the Thompson Center Arms firearms.
In the good "Old Contenders", I have found that military surplus ammo seems to have extra tough primers, and for most cases will cause many misfires, or the need to hit twice to get them to shoot. Or in some cases will not fire at all with the Contenders. With G2's and Encore's, your odds seem to be better.
As for commercial primers, I have found that CCI have the thickest cups. Federal seem to have the thinnest cups. All others seem to fall somewhere in the middle. That is why I choose Federal for all my fire forming of the brass, then switch to a primer of choice after.
Excessive headspace seems to be the biggest reason for misfires. But is by far not the only cause. If your headspace is correct, or near to correct, and you are having misfires, change your brand of primers and see it that stops your problem.
In some caliber's, the choice of primers seems to have a big effect on velocity and accuracy, on others is seems to have little to no effect on either.