@survivor and joebanana there is no way that is the printers fault. If anything it’s the printers fault for telling the customer that there is a chance (good one albeit) that the border may be uneven and completely gone on some cards. And for what cutting variances may come to as long as they disclose it to you and you are away if it’s in their tolerances then you really can’t complain. Now if they said sure no problem we can do that, and it will be just like it looks, then it’s their fault. If you want to do a border like that it has to be more than just the 6px that you supplied to be sure unless you want to pay for the extra care in cutting stage.
As quickandeasy says you get what your pay for, but there are good printers out there, just need to look for them.
Business card borders should be at least 0.25”. Pixel borders are virtually impossible for cutters unless its a laser cutter, which most printers do not use. You will need the first 0.125” then the 0.25” border after that. There will still be movement during the cutting process, but because the border is so thick – the difference in size on each side is negligible. A 6 pixel border is waaaaaaaaaaaay too thin.