Once you have your paper punch apart, you can check the edges of the base and the punch blade, to see if there are any rough patches of metal on the blade or the base, which are causing the punch to stick. Punches are machined with a very high tolerance, so they must fit exactly through the base to punch out the paper design. slight roughness on the edges of either the blade, or the sections of the base the blades pass through, can cause the blade to stick and the punch to seize, even on regular printer weight paper. Note: Craft punches, especially elaborate ones, are not designed to cut through lightweight card. More elaborate punches may only work with lighter weghts of printer paper. Tips on Paper for Printable Miniatures discusses the various weights of paper and how to store it so it won't jam in a printer. The same information is useful for working with decorative paper punches.
If Your Punch Has Rough Edges - if you can see rough edges on the cutting parts of your punch, or you can see bits of metal that might cause the punch to bind, use very fine sandpaper (600 grit or finer) , or a fine needle file to clear away the excess metal, or file it smooth. Work carefully, you don't want to change or dull the blades of your paper punch! Do not work a file or sandpaper across the sharp edges, only file the sides of the cutting blades to remove rough spots. Filing or sanding across the edges will dull the blade.