Do not cut straight across
Make sure you are using one of those smaller, rounded-blade clippers; they are the ones designed for fingers before clipping your nails, and they work great on your smaller toes. The big, straight-blade nail clipper in your dopp kit is for your big toe.
When using the rounded fingernail clippers, you still need to cut the nails at an angle, since the curve of the blade does not match the curve of your nail perfectly. A common mistake men and women make when trimming their own nails is the angle at which they approach it, clipping straight-on can bend and ultimately ruin the nail. Instead, you should be clipping the nail three or four times as you travel across the top.